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<title>Network science</title>

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Network science

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Network_(disambiguation)" class="mw-redirect mw-disambig" title="Network (disambiguation)">Network (disambiguation)</a>.</div>

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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle" style="padding-bottom:0.15em;">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle" style="font-size:175%;"></th></tr><tr><td class="sidebar-image"><div class="center"><div class="center">

</div>

</div></div></td></tr><tr><th class="sidebar-heading">

<div class="hlist"><ul><li><a href="Network_theory" title="Network theory">Theory</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;">

<ul><li><a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">Graph</a></li>

<li><a href="Complex_network" title="Complex network">Complex network</a></li>

<li><a href="Complex_contagion" title="Complex contagion">Contagion</a></li>

<li><a href="Small-world_network" title="Small-world network">Small-world</a></li>

<li><a href="Scale-free_network" title="Scale-free network">Scale-free</a></li>

<li><a href="Community_structure" title="Community structure">Community structure</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Evolving_networks" class="mw-redirect" title="Evolving networks">Evolution</a></li>

<li><a href="Network_controllability" title="Network controllability">Controllability</a></li>

<li><a href="Graph_drawing" title="Graph drawing">Graph drawing</a></li>

<li><a href="Social_capital" title="Social capital">Social capital</a></li>

<li><a href="Link_analysis" title="Link analysis">Link analysis</a></li>

<li><a href="Combinatorial_optimization" title="Combinatorial optimization">Optimization</a></li>

<li><a href="Reciprocity_(network_science)" title="Reciprocity (network science)">Reciprocity</a></li>

<li><a href="Triadic_closure" title="Triadic closure">Closure</a></li>

<li><a href="Homophily" title="Homophily">Homophily</a></li>

<li><a href="Transitive_relation" title="Transitive relation">Transitivity</a></li>

<li><a href="Preferential_attachment" title="Preferential attachment">Preferential attachment</a></li>

<li><a href="Balance_theory" title="Balance theory">Balance theory</a></li>

<li><a href="Network_effect" title="Network effect">Network effect</a></li>

<li><a href="Social_influence" title="Social influence">Social influence</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Network types</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;">

<ul><li><a href="Computer_network" title="Computer network">Informational (computing)</a></li>

<li><a href="Telecommunications_network" title="Telecommunications network">Telecommunication</a></li>

<li><a href="Transport_network" class="mw-redirect" title="Transport network">Transport</a></li>

<li><a href="Social_network" title="Social network">Social</a></li>

<li><a href="Scientific_collaboration_network" title="Scientific collaboration network">Scientific collaboration</a></li>

<li><a href="Biological_network" title="Biological network">Biological</a></li>

<li><a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural</a></li>

<li><a href="Interdependent_networks" title="Interdependent networks">Interdependent</a></li>

<li><a href="Semantic_network" title="Semantic network">Semantic</a></li>

<li><a href="Spatial_network" title="Spatial network">Spatial</a></li>

<li><a href="Dependency_network" title="Dependency network">Dependency</a></li>

</tr><tr><th class="sidebar-heading">

<a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">Graphs</a></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;">

Features</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Clique_(graph_theory)" title="Clique (graph theory)">Clique</a></li>

<li><a href="Connected_component_(graph_theory)" class="mw-redirect" title="Connected component (graph theory)">Component</a></li>

<li><a href="Cycle_(graph_theory)" title="Cycle (graph theory)">Cycle</a></li>

<li><a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">Data structure</a></li>

<li><a href="Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">Neighborhood</a></li>

<li><a href="Vertex_(graph_theory)" title="Vertex (graph theory)">Vertex</a></li>

<li><a href="Adjacency_list" title="Adjacency list">Adjacency list</a> / <a href="Adjacency_matrix" title="Adjacency matrix">matrix</a></li>

<li><a href="Incidence_list" class="mw-redirect" title="Incidence list">Incidence list</a> / <a href="Incidence_matrix" title="Incidence matrix">matrix</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;">

Types</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Bipartite_graph" title="Bipartite graph">Bipartite</a></li>

<li><a href="Complete_graph" title="Complete graph">Complete</a></li>

<li><a href="Directed_graph" title="Directed graph">Directed</a></li>

<li><a href="Hypergraph" title="Hypergraph">Hyper</a></li>

<li><a href="Labeled_graph" class="mw-redirect" title="Labeled graph">Labeled</a></li>

<li><a href="Multigraph" title="Multigraph">Multi</a></li>

<li><a href="Random_graph" title="Random graph">Random</a></li>

<li><a href="Weighted_graph" class="mw-redirect" title="Weighted graph">Weighted</a></li></ul></td>

</tr></tbody></table></td>

</tr><tr><th class="sidebar-heading">

<div class="hlist"><ul><li><a href="Metrics_(networking)" title="Metrics (networking)">Metrics</a></li><li><a href="List_of_algorithms#Networking" title="List of algorithms">Algorithms</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;">

<ul><li><a href="Centrality" title="Centrality">Centrality</a></li>

<li><a href="Degree_(graph_theory)" title="Degree (graph theory)">Degree</a></li>

<li><a href="Network_motif" title="Network motif">Motif</a></li>

<li><a href="Clustering_coefficient" title="Clustering coefficient">Clustering</a></li>

<li><a href="Degree_distribution" title="Degree distribution">Degree distribution</a></li>

<li><a href="Assortativity" title="Assortativity">Assortativity</a></li>

<li><a href="Distance_(graph_theory)" title="Distance (graph theory)">Distance</a></li>

<li><a href="Modularity_(networks)" title="Modularity (networks)">Modularity</a></li>

<li><a href="Efficiency_(network_science)" title="Efficiency (network science)">Efficiency</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Models</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;">

Topology</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Random_graph" title="Random graph">Random graph</a></li>

<li><a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi</a></li>

<li><a href="Barab%C3%A1si%E2%80%93Albert_model" title="Barabási–Albert model">Barabási–Albert</a></li>

<li><a href="Bianconi%E2%80%93Barab%C3%A1si_model" title="Bianconi–Barabási model">Bianconi–Barabási</a></li>

<li><a href="Fitness_model_(network_theory)" title="Fitness model (network theory)">Fitness model</a></li>

<li><a href="Watts%E2%80%93Strogatz_model" title="Watts–Strogatz model">Watts–Strogatz</a></li>

<li><a href="Exponential_random_graph_models" class="mw-redirect" title="Exponential random graph models">Exponential random (ERGM)</a></li>

<li><a href="Random_geometric_graph" title="Random geometric graph">Random geometric (RGG)</a></li>

<li><a href="Hyperbolic_geometric_graph" title="Hyperbolic geometric graph">Hyperbolic (HGN)</a></li>

<li><a href="Hierarchical_network_model" title="Hierarchical network model">Hierarchical</a></li>

<li><a href="Stochastic_block_model" title="Stochastic block model">Stochastic block</a></li>

<li><a href="Blockmodeling" title="Blockmodeling">Blockmodeling</a></li>

<li><a href="Maximum-entropy_random_graph_model" title="Maximum-entropy random graph model">Maximum entropy</a></li>

<li><a href="Soft_configuration_model" title="Soft configuration model">Soft configuration</a></li>

<li><a href="Lancichinetti%E2%80%93Fortunato%E2%80%93Radicchi_benchmark" title="Lancichinetti–Fortunato–Radicchi benchmark">LFR Benchmark</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;">

Dynamics</th></tr><tr><td class="sidebar-content">

<ul><li><a href="Boolean_network" title="Boolean network">Boolean network</a></li>

<li><a href="Agent-based_model" title="Agent-based model">agent based</a></li>

<li><a href="Epidemic_model" class="mw-redirect" title="Epidemic model">Epidemic</a>/<a href="SIR_model" class="mw-redirect" title="SIR model">SIR</a></li></ul></td>

</tr></tbody></table></td>

</tr><tr><th class="sidebar-heading">

<div class="hlist"><ul><li>Lists</li><li>Categories</li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;">

<ul><li><a href="List_of_network_theory_topics" title="List of network theory topics">Topics</a></li>

<li><a href="Social_network_analysis_software" title="Social network analysis software">Software</a></li>

<li><a href="List_of_network_scientists" title="List of network scientists">Network scientists</a></li></ul>

<ul><li>Category:Network theory</li>

<li>Category:Graph theory</li></ul></td>

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<table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="Complex_system" title="Complex system">Complex systems</a></th></tr><tr><th class="sidebar-heading">

Topics</th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Self-organization" title="Self-organization">Self-organization</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="Emergence" title="Emergence">Emergence</a> </div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Collective_behavior" title="Collective behavior">Collective behavior</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="Social_dynamics" title="Social dynamics">Social dynamics</a>

<a href="Collective_intelligence" title="Collective intelligence">Collective intelligence</a>

<a href="Collective_action" title="Collective action">Collective action</a>

<a href="Self-organized_criticality" title="Self-organized criticality">Self-organized criticality</a>

<a href="Herd_mentality" title="Herd mentality">Herd mentality</a>

<a href="Phase_transition" title="Phase transition">Phase transition</a>

<a href="Agent-based_model" title="Agent-based model">Agent-based modelling</a>

<a href="Synchronization" title="Synchronization">Synchronization</a>

<a href="Ant_colony_optimization_algorithms" title="Ant colony optimization algorithms">Ant colony optimization</a>

<a href="Particle_swarm_optimization" title="Particle swarm optimization">Particle swarm optimization</a>

<a href="Swarm_behaviour" title="Swarm behaviour">Swarm behaviour</a>

<a href="Collective_consciousness" title="Collective consciousness">Collective consciousness</a> </div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"></div><div class="sidebar-list-content mw-collapsible-content"><a href="Scale-free_network" title="Scale-free network">Scale-free networks</a>

<a href="Social_network_analysis" title="Social network analysis">Social network analysis</a>

<a href="Small-world_network" title="Small-world network">Small-world networks</a>

<a href="Centrality" title="Centrality">Centrality</a>

<a href="Network_motif" title="Network motif">Motifs</a>

<a href="Graph_theory" title="Graph theory">Graph theory</a>

<a href="Scalability" title="Scalability">Scaling</a>

<a href="Robustness_(computer_science)" title="Robustness (computer science)">Robustness</a>

<a href="Systems_biology" title="Systems biology">Systems biology</a>

<a href="Dynamic_network_analysis" title="Dynamic network analysis">Dynamic networks</a>

<a href="Complex_adaptive_system" title="Complex adaptive system">Adaptive networks</a></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Evolution" title="Evolution">Evolution</a> and <a href="Adaptation" title="Adaptation">adaptation</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">Artificial neural network</a>

<a href="Evolutionary_computation" title="Evolutionary computation">Evolutionary computation</a>

<a href="Genetic_algorithm" title="Genetic algorithm">Genetic algorithms</a>

<a href="Genetic_programming" title="Genetic programming">Genetic programming</a>

<a href="Artificial_life" title="Artificial life">Artificial life</a>

<a href="Machine_learning" title="Machine learning">Machine learning</a>

<a href="Evolutionary_developmental_biology" title="Evolutionary developmental biology">Evolutionary developmental biology</a>

<a href="Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a>

<a href="Evolutionary_robotics" title="Evolutionary robotics">Evolutionary robotics</a>

<a href="Evolvability" title="Evolvability">Evolvability</a></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Pattern_formation" title="Pattern formation">Pattern formation</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="Fractal" title="Fractal">Fractals</a>

<a href="Reaction%E2%80%93diffusion_system" title="Reaction–diffusion system">Reaction–diffusion systems</a>

<a href="Partial_differential_equation" title="Partial differential equation">Partial differential equations</a>

<a href="Dissipative_system" title="Dissipative system">Dissipative structures</a>

<a href="Percolation" title="Percolation">Percolation</a>

<a href="Cellular_automaton" title="Cellular automaton">Cellular automata</a>

<a href="Spatial_ecology" title="Spatial ecology">Spatial ecology</a>

<a href="Self-replication" title="Self-replication">Self-replication</a>

<a href="Geomorphology" title="Geomorphology">Geomorphology</a></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Systems_theory" title="Systems theory">Systems theory</a> and <a href="Cybernetics" title="Cybernetics">cybernetics</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="Autopoiesis" title="Autopoiesis">Autopoiesis</a>

<a href="Conversation_theory" title="Conversation theory">Conversation theory</a>

<a href="Entropy" title="Entropy">Entropy</a>

<a href="Feedback" title="Feedback">Feedback</a>

<a href="Goal_orientation" title="Goal orientation">Goal-oriented</a>

<a href="Homeostasis" title="Homeostasis">Homeostasis</a>

<a href="Information_theory" title="Information theory">Information theory</a>

<a href="Operationalization" title="Operationalization">Operationalization</a>

<a href="Second-order_cybernetics" title="Second-order cybernetics">Second-order cybernetics</a>

<a href="Self-reference" title="Self-reference">Self-reference</a>

<a href="System_dynamics" title="System dynamics">System dynamics</a>

<a href="Systems_science" title="Systems science">Systems science</a>

<a href="Systems_thinking" title="Systems thinking">Systems thinking</a>

<a href="Sensemaking" title="Sensemaking">Sensemaking</a>

<a href="Variety_(cybernetics)" title="Variety (cybernetics)">Variety</a>

<a href="Theory_of_computation" title="Theory of computation">Theory of computation</a> </div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Nonlinear_dynamics" class="mw-redirect" title="Nonlinear dynamics">Nonlinear dynamics</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="Time_series" title="Time series">Time series analysis</a>

<a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a>

<a href="Phase_space" title="Phase space">Phase space</a>

<a href="Attractor" title="Attractor">Attractors</a>

<a href="Population_dynamics" title="Population dynamics">Population dynamics</a>

<a href="Chaos_theory" title="Chaos theory">Chaos</a>

<a href="Multistability" title="Multistability">Multistability</a>

<a href="Bifurcation_theory" title="Bifurcation theory">Bifurcation</a>

<a href="Coupled_map_lattice" title="Coupled map lattice">Coupled map lattices</a></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Game_theory" title="Game theory">Game theory</a></div><div class="sidebar-list-content mw-collapsible-content"><a href="Prisoner's_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a>

<a href="Rational_choice_model" title="Rational choice model">Rational choice theory</a>

<a href="Bounded_rationality" title="Bounded rationality">Bounded rationality</a>

<a href="Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a> </div></div></td>

</tr><tr><td class="sidebar-navbar"></td></tr></tbody></table>

<table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-top-image"></td></tr><tr><th class="sidebar-title">Information mapping</th></tr><tr><th class="sidebar-heading" style="background:transparent;">

Topics and fields</th></tr><tr><td class="sidebar-content" style="padding-bottom:0.9em;">

<ul><li><a href="Business_decision_mapping" title="Business decision mapping">Business decision mapping</a></li>

<li><a href="Data_and_information_visualization" title="Data and information visualization">Data visualization</a></li>

<li><a href="Graphic_communication" title="Graphic communication">Graphic communication</a></li>

<li><a href="Infographic" title="Infographic">Infographics</a></li>

<li><a href="Information_design" title="Information design">Information design</a></li>

<li><a href="Knowledge_visualization" class="mw-redirect" title="Knowledge visualization">Knowledge visualization</a></li>

<li><a href="Mental_model" title="Mental model">Mental model</a></li>

<li><a href="Morphological_analysis_(problem-solving)" title="Morphological analysis (problem-solving)">Morphological analysis</a></li>

<li><a href="Ontology_(information_science)" title="Ontology (information science)">Ontology (information science)</a></li>

<li><a href="Schema_(psychology)" title="Schema (psychology)">Schema (psychology)</a></li>

<li><a href="Visual_analytics" title="Visual analytics">Visual analytics</a></li>

<li><a href="Visual_language" title="Visual language">Visual language</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:transparent;">

Node–link approaches</th></tr><tr><td class="sidebar-content" style="padding-bottom:0.9em;">

<ul><li><a href="Argument_map" title="Argument map">Argument map</a></li>

<li><a href="Cladistics" title="Cladistics">Cladistics</a></li>

<li><a href="Cognitive_map" title="Cognitive map">Cognitive map</a></li>

<li><a href="Concept_lattice" class="mw-redirect" title="Concept lattice">Concept lattice</a></li>

<li><a href="Concept_map" title="Concept map">Concept map</a></li>

<li><a href="Conceptual_graph" title="Conceptual graph">Conceptual graph</a></li>

<li><a href="Decision_tree" title="Decision tree">Decision tree</a></li>

<li><a href="Dendrogram" title="Dendrogram">Dendrogram</a></li>

<li><a href="Graph_drawing" title="Graph drawing">Graph drawing</a></li>

<li><a href="Hyperbolic_tree" title="Hyperbolic tree">Hyperbolic tree</a></li>

<li><a href="Hypertext" title="Hypertext">Hypertext</a></li>

<li><a href="Issue-based_information_system" title="Issue-based information system">Issue map</a></li>

<li><a href="Issue_tree" title="Issue tree">Issue tree</a></li>

<li><a href="Layered_graph_drawing" title="Layered graph drawing">Layered graph drawing</a></li>

<li><a href="Object-role_modeling" class="mw-redirect" title="Object-role modeling">Object-role modeling</a></li>

<li><a href="Organizational_chart" title="Organizational chart">Organizational chart</a></li>

<li><a href="Pathfinder_network" title="Pathfinder network">Pathfinder network</a></li>

<li><a href="Radial_tree" title="Radial tree">Radial tree</a></li>

<li><a href="Semantic_network" title="Semantic network">Semantic network</a></li>

<li><a href="Sociogram" title="Sociogram">Sociogram</a></li>

<li><a href="Timeline" title="Timeline">Timeline</a></li>

<li><a href="Topic_map" title="Topic map">Topic map</a></li>

<li><a href="Tree_structure" title="Tree structure">Tree structure</a></li>

<li><a href="ZigZag_(software)" title="ZigZag (software)">ZigZag</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:transparent;">

See also</th></tr><tr><td class="sidebar-content" style="padding-bottom:0.9em;">

<ul><li><a href="Design_rationale" title="Design rationale">Design rationale</a></li>

<li><a href="Diagrammatic_reasoning" title="Diagrammatic reasoning">Diagrammatic reasoning</a></li>

<li><a href="Entity%E2%80%93relationship_model" title="Entity–relationship model">Entity–relationship model</a></li>

<li><a href="Geovisualization" title="Geovisualization">Geovisualization</a></li>

<li><a href="List_of_concept-_and_mind-mapping_software" title="List of concept- and mind-mapping software">List of concept- and mind-mapping software</a></li>

<li><a href="Ontology" title="Ontology">Ontology (philosophy)</a></li>

<li><a href="Problem_structuring_methods" title="Problem structuring methods">Problem structuring methods</a></li>

<li><a href="Semantic_Web" title="Semantic Web">Semantic Web</a></li>

<li><a href="Treemapping" title="Treemapping">Treemapping</a></li>

<li><a href="Wicked_problem" title="Wicked problem">Wicked problem</a></li></ul></td>

</tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;"></td></tr></tbody></table>

Network science is an academic field which studies <a href="Complex_network" title="Complex network">complex networks</a> such as <a href="Telecommunication_network" class="mw-redirect" title="Telecommunication network">telecommunication networks</a>, <a href="Computer_network" title="Computer network">computer networks</a>, <a href="Biological_network" title="Biological network">biological networks</a>, <a href="Cognitive_network" title="Cognitive network">cognitive</a> and <a href="Semantic_network" title="Semantic network">semantic networks</a>, and <a href="Social_network" title="Social network">social networks</a>, considering distinct elements or actors represented by nodes (or vertices) and the connections between the elements or actors as links (or edges). The field draws on theories and methods including <a href="Graph_theory" title="Graph theory">graph theory</a> from mathematics, <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> from physics, <a href="Data_mining" title="Data mining">data mining</a> and <a href="Information_visualization" class="mw-redirect" title="Information visualization">information visualization</a> from computer science, <a href="Inferential_statistics" class="mw-redirect" title="Inferential statistics">inferential modeling</a> from statistics, and <a href="Social_structure" title="Social structure">social structure</a> from sociology. The <a href="United_States_National_Research_Council" class="mw-redirect" title="United States National Research Council">United States National Research Council</a> defines network science as "the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena."<a href="#cite_note-NRC-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Background_and_history">Background and history</h2></div>

The study of networks has emerged in diverse disciplines as a means of analyzing complex relational data. The earliest known paper in this field is the famous <a href="Seven_Bridges_of_K%C3%B6nigsberg" title="Seven Bridges of Königsberg">Seven Bridges of Königsberg</a> written by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in 1736. Euler's mathematical description of vertices and edges was the foundation of <a href="Graph_theory" title="Graph theory">graph theory</a>, a branch of mathematics that studies the properties of pairwise relations in a network structure. The field of <a href="Graph_theory" title="Graph theory">graph theory</a> continued to develop and found applications in chemistry (Sylvester, 1878).

<a href="D%C3%A9nes_K%C5%91nig" title="Dénes Kőnig">Dénes Kőnig</a>, a Hungarian mathematician and professor, wrote the first book in Graph Theory, entitled "Theory of finite and infinite graphs", in 1936.<a href="#cite_note-2">[2]</a>

In the 1930s <a href="Jacob_Moreno" class="mw-redirect" title="Jacob Moreno">Jacob Moreno</a>, a psychologist in the <a href="Gestalt_psychology" title="Gestalt psychology">Gestalt</a> tradition, arrived in the United States. He developed the <a href="Sociogram" title="Sociogram">sociogram</a> and presented it to the public in April 1933 at a convention of medical scholars. Moreno claimed that "before the advent of sociometry no one knew what the interpersonal structure of a group 'precisely' looked like".<a href="#cite_note-3">[3]</a> The sociogram was a representation of the social structure of a group of elementary school students. The boys were friends of boys and the girls were friends of girls with the exception of one boy who said he liked a single girl. The feeling was not reciprocated. This network representation of social structure was found so intriguing that it was printed in <a href="The_New_York_Times" title="The New York Times">The New York Times</a>.<a href="#cite_note-4">[4]</a> The sociogram has found many applications and has grown into the field of <a href="Social_network_analysis" title="Social network analysis">social network analysis</a>.

Probabilistic theory in network science developed as an offshoot of <a href="Graph_theory" title="Graph theory">graph theory</a> with <a href="Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> and <a href="Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Alfréd Rényi</a>'s eight famous papers on <a href="Random_graphs" class="mw-redirect" title="Random graphs">random graphs</a>. For <a href="Social_networks" class="mw-redirect" title="Social networks">social networks</a> the <a href="Exponential_random_graph_model" class="mw-redirect" title="Exponential random graph model">exponential random graph model</a> or p* is a notational framework used to represent the probability space of a tie occurring in a <a href="Social_network" title="Social network">social network</a>. An alternate approach to network probability structures is the <a href="Network_probability_matrix" title="Network probability matrix">network probability matrix</a>, which models the probability of edges occurring in a network, based on the historic presence or absence of the edge in a sample of networks.

Interest in networks exploded around 2000, following new discoveries that offered novel mathematical framework to describe different network topologies, leading to the term 'network science'. <a href="Albert-L%C3%A1szl%C3%B3_Barab%C3%A1si" title="Albert-László Barabási">Albert-László Barabási</a> and <a href="Reka_Albert" class="mw-redirect" title="Reka Albert">Reka Albert</a> discovered the <a href="Scale-free_network" title="Scale-free network">scale-free networks</a><a href="#cite_note-:2-5">[5]</a> nature of many real networks, from the WWW to the cell. The scale-free property captures the fact that in real network hubs coexist with many small degree vertices, and the authors offered a dynamical model to explain the origin of this scale-free state.<a href="#cite_note-:2-5">[5]</a> <a href="Duncan_J._Watts" title="Duncan J. Watts">Duncan Watts</a> and <a href="Steven_Strogatz" title="Steven Strogatz">Steven Strogatz</a> reconciled empirical data on networks with mathematical representation, describing the <a href="Small-world_network" title="Small-world network">small-world network</a>.<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Network_Classification">Network Classification</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Deterministic_Network">Deterministic Network</h3></div>

The definition of deterministic network is defined compared with the definition of probabilistic network. In un-weighted deterministic networks, edges either exist or not, usually we use 0 to represent non-existence of an edge while 1 to represent existence of an edge. In weighted deterministic networks, the edge value represents the weight of each edge, for example, the strength level.

<div class="mw-heading mw-heading3"><h3 id="Probabilistic_Network">Probabilistic Network</h3></div>

In probabilistic networks, values behind each edge represent the likelihood of the existence of each edge. For example, if one edge has a value equals to 0.9, we say the existence probability of this edge is 0.9.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Network_properties">Network properties</h2></div>

Often, networks have certain attributes that can be calculated to analyze the properties & characteristics of the network. The behavior of these network properties often define <a href="Network_model" title="Network model">network models</a> and can be used to analyze how certain models contrast to each other. Many of the definitions for other terms used in network science can be found in <a href="Glossary_of_graph_theory" title="Glossary of graph theory">Glossary of graph theory</a>.

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Network_homophily" title="Network homophily">Network homophily</a></div>

The size of a network can refer to the number of nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> or, less commonly, the number of edges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> which (for connected graphs with no multi-edges) can range from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N-1}</annotation>

</semantics>

</math><img src="./86aeb216b214f70df1341f34ce273cd3582ce2aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N-1}" loading="lazy"> (a tree) to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\max }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\max }}</annotation>

</semantics>

</math><img src="./c216e15937a36c6d77e18204a4ade1162daa93bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.006ex; height:2.509ex;" alt="{\displaystyle E_{\max }}" loading="lazy"> (a complete graph). In the case of a simple graph (a network in which at most one (undirected) edge exists between each pair of vertices, and in which no vertices connect to themselves), we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\max }={\tbinom {N}{2}}=N(N-1)/2}">

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\max }={\tbinom {N}{2}}=N(N-1)/2}</annotation>

</semantics>

</math><img src="./c1494a8b696d03cd0b444d78fc62d5fa9b1b6ebb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.057ex; height:3.343ex;" alt="{\displaystyle E_{\max }={\tbinom {N}{2}}=N(N-1)/2}" loading="lazy">; for directed graphs (with no self-connected nodes), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\max }=N(N-1)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\max }=N(N-1)}</annotation>

</semantics>

</math><img src="./d2da7cb2832a96eba12231c94d944218e1423b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.044ex; height:2.843ex;" alt="{\displaystyle E_{\max }=N(N-1)}" loading="lazy">; for directed graphs with self-connections allowed, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\max }=N^{2}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\max }=N^{2}}</annotation>

</semantics>

</math><img src="./34446df16c764eadea7c1c884a023e31612d950d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.282ex; height:3.009ex;" alt="{\displaystyle E_{\max }=N^{2}}" loading="lazy">. In the circumstance of a graph within which multiple edges may exist between a pair of vertices, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\max }=\infty }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\max }=\infty }</annotation>

</semantics>

</math><img src="./78611d4fe42c972b618ee81535fe0a986913a3b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.429ex; height:2.509ex;" alt="{\displaystyle E_{\max }=\infty }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Density">Density</h3></div>

The density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> of a network is defined as a normalized ratio between 0 and 1 of the number of edges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> to the number of possible edges in a network with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> nodes. Network density is a measure of the percentage of "optional" edges that exist in the network and can be computed as

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D={\frac {E-E_{\mathrm {min} }}{E_{\mathrm {max} }-E_{\mathrm {min} }}}}</annotation>

</semantics>

</math><img src="./ea0abbca28a9bb5aab449ebad8a9e2464ac9769e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.393ex; height:5.509ex;" alt="{\displaystyle D={\frac {E-E_{\mathrm {min} }}{E_{\mathrm {max} }-E_{\mathrm {min} }}}}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {min} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {min} }}</annotation>

</semantics>

</math><img src="./c05ade5f06af1310eb1b6f32ec4fa7d95947df29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.688ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {min} }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {max} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {max} }}</annotation>

</semantics>

</math><img src="./87165fb05bd3e9b74c727284081c8980ab2e05a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.006ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {max} }}" loading="lazy"> are the minimum and maximum number of edges in a connected network with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> nodes, respectively. In the case of simple graphs, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {max} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {max} }}</annotation>

</semantics>

</math><img src="./87165fb05bd3e9b74c727284081c8980ab2e05a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.006ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {max} }}" loading="lazy"> is given by the <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {N}{2}}}">

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tbinom {N}{2}}}</annotation>

</semantics>

</math><img src="./b452d1fbc30268e85e2867a43c0a88a362b3e253.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.589ex; height:3.343ex;" alt="{\displaystyle {\tbinom {N}{2}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {min} }=N-1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {min} }=N-1}</annotation>

</semantics>

</math><img src="./27863ecd778f60ebd2a545ce1a70d7bf36fbbde9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.853ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {min} }=N-1}" loading="lazy">, giving density

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D={\frac {E-(N-1)}{E_{\mathrm {max} }-(N-1)}}={\frac {2(E-N+1)}{N(N-3)+2}}}</annotation>

</semantics>

</math><img src="./28394b3677d1d7d35e2dec7691c10f29d04118e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.458ex; height:6.509ex;" alt="{\displaystyle D={\frac {E-(N-1)}{E_{\mathrm {max} }-(N-1)}}={\frac {2(E-N+1)}{N(N-3)+2}}}" loading="lazy">.

Another possible equation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\frac {T-2N+2}{N(N-3)+2}},}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D={\frac {T-2N+2}{N(N-3)+2}},}</annotation>

</semantics>

</math><img src="./dffe65f1e8ab9923564b46279f2d62741ebf4097.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.448ex; height:6.009ex;" alt="{\displaystyle D={\frac {T-2N+2}{N(N-3)+2}},}" loading="lazy"> whereas the ties <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> are unidirectional (Wasserman & Faust 1994).<a href="#cite_note-8">[8]</a> This gives a better overview over the network density, because unidirectional relationships can be measured.

<div class="mw-heading mw-heading3"><h3 id="Planar_network_density">Planar network density</h3></div>

The density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> of a network, where there is no intersection between edges, is defined as a ratio of the number of edges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> to the number of possible edges in a network with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> nodes, given by a graph with no intersecting edges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{\max }=3N-6)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E_{\max }=3N-6)}</annotation>

</semantics>

</math><img src="./3b28ee16bdd1de0c2ba19fc9871bef79b7c58bfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.143ex; height:2.843ex;" alt="{\displaystyle (E_{\max }=3N-6)}" loading="lazy">, giving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\frac {E-N+1}{2N-5}}.}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D={\frac {E-N+1}{2N-5}}.}</annotation>

</semantics>

</math><img src="./075e2d24b7f8d80e412a331a723350389fbf69b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.188ex; height:5.343ex;" alt="{\displaystyle D={\frac {E-N+1}{2N-5}}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Average_degree">Average degree</h3></div>

The <a href="Degree_(graph_theory)" title="Degree (graph theory)">degree</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of a node is the number of edges connected to it. Closely related to the density of a network is the average degree, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle ={\tfrac {2E}{N}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle k\rangle ={\tfrac {2E}{N}}}</annotation>

</semantics>

</math><img src="./f91662782211f80643a25f3b9a791c32f91b5eba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.033ex; height:3.509ex;" alt="{\displaystyle \langle k\rangle ={\tfrac {2E}{N}}}" loading="lazy"> (or, in the case of directed graphs, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle ={\tfrac {E}{N}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle k\rangle ={\tfrac {E}{N}}}</annotation>

</semantics>

</math><img src="./11595c48e5d9ddf66cebef7f2c4c71acfea2860f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.414ex; height:3.509ex;" alt="{\displaystyle \langle k\rangle ={\tfrac {E}{N}}}" loading="lazy">, the former factor of 2 arising from each edge in an undirected graph contributing to the degree of two distinct vertices). In the <a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">ER random graph model</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(N,p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(N,p)}</annotation>

</semantics>

</math><img src="./d6a9303ccd0a3fffea87321e323a9ebbff8e7cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.903ex; height:2.843ex;" alt="{\displaystyle G(N,p)}" loading="lazy">) we can compute the expected value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle k\rangle }</annotation>

</semantics>

</math><img src="./79c8e81fd47c64b42f310aa18c5197183dcbb0d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.021ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle }" loading="lazy"> (equal to the expected value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of an arbitrary vertex): a random vertex has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N-1}</annotation>

</semantics>

</math><img src="./86aeb216b214f70df1341f34ce273cd3582ce2aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N-1}" loading="lazy"> other vertices in the network available, and with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, connects to each. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [\langle k\rangle ]=\mathbb {E} [k]=p(N-1)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [\langle k\rangle ]=\mathbb {E} [k]=p(N-1)}</annotation>

</semantics>

</math><img src="./e39043016f1e04a734b4d1253e1ad7301987a4ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.162ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [\langle k\rangle ]=\mathbb {E} [k]=p(N-1)}" loading="lazy">.

<a href="Degree_distribution" title="Degree distribution"><big>Degree distribution</big></a>

The degree distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(k)}</annotation>

</semantics>

</math><img src="./b41614fb84549b21f2c7f2793bbd8a87a2105027.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle P(k)}" loading="lazy"> is a fundamental property of both real networks, such as the <a href="Internet" title="Internet">Internet</a> and <a href="Social_networks" class="mw-redirect" title="Social networks">social networks</a>, and of theoretical models. The degree distribution P(k) of a network is defined to be the fraction of nodes in the network with degree k. The simplest network model, for example, the (Erdős–Rényi model) <a href="Random_graph" title="Random graph">random graph</a>, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a <a href="Binomial_distribution" title="Binomial distribution">binomial distribution</a> of degrees k (or <a href="Poisson_distribution" title="Poisson distribution">Poisson</a> in the limit of large n). Most real networks, from the <a href="World_Wide_Web" title="World Wide Web">WWW</a> to <a href="Interactome" title="Interactome">protein interaction networks</a>, however, have a degree distribution that are highly <a href="Skewness" title="Skewness">right-skewed</a>, meaning that a large majority of nodes have low degree but a small number, known as "hubs", have high degree. For such <a href="Scale-free_network" title="Scale-free network">scale-free networks</a> the degree distribution approximately follows a <a href="Power_law" title="Power law">power law</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k)\sim k^{-\gamma }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(k)\sim k^{-\gamma }}</annotation>

</semantics>

</math><img src="./fd197472712748ee93f163b2832421657a265d34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.479ex; height:3.009ex;" alt="{\displaystyle P(k)\sim k^{-\gamma }}" loading="lazy">, where γ is the degree exponent, and is a constant. Such <a href="Scale-free_networks" class="mw-redirect" title="Scale-free networks">scale-free networks</a> have unexpected structural and dynamical properties, rooted in the diverging second moment of the degree distribution. <a href="#cite_note-BA-9">[9]</a><a href="#cite_note-AB-10">[10]</a><a href="#cite_note-Doro-11">[11]</a><a href="#cite_note-PSY-12">[12]</a>

<div class="mw-heading mw-heading3"><h3 id="Average_shortest_path_length_(or_characteristic_path_length)">Average shortest path length (or characteristic path length)</h3></div>

The average shortest path length is calculated by finding the <a href="Shortest_path" class="mw-redirect" title="Shortest path">shortest path</a> between all pairs of nodes, and taking the average over all paths of the length thereof (the length being the number of intermediate edges contained in the path, i.e., the distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{u,v}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{u,v}}</annotation>

</semantics>

</math><img src="./02bbfeee569a787de5aac5b97089e0d30221b4f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.636ex; height:2.843ex;" alt="{\displaystyle d_{u,v}}" loading="lazy"> between the two vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u,v}</annotation>

</semantics>

</math><img src="./7e66f4b32a0181923cc1337a5634f38241e5c697.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle u,v}" loading="lazy"> within the graph). This shows us, on average, the number of steps it takes to get from one member of the network to another. The behavior of the expected average shortest path length (that is, the ensemble average of the average shortest path length) as a function of the number of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> of a random network model defines whether that model exhibits the small-world effect; if it scales as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\ln N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(\ln N)}</annotation>

</semantics>

</math><img src="./3aece3e998ff5c7451f9ee3f85a55323ad0fa94e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.973ex; height:2.843ex;" alt="{\displaystyle O(\ln N)}" loading="lazy">, the model generates small-world nets. For faster-than-logarithmic growth, the model does not produce small worlds. The special case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\ln \ln N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(\ln \ln N)}</annotation>

</semantics>

</math><img src="./10176e6ea41625d2fd810143c787c1b270a47660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.299ex; height:2.843ex;" alt="{\displaystyle O(\ln \ln N)}" loading="lazy"> is known as ultra-small world effect.

<div class="mw-heading mw-heading3"><h3 id="Diameter_of_a_network">Diameter of a network</h3></div>

As another means of measuring network graphs, we can define the diameter of a network as the longest of all the calculated shortest paths in a network. It is the shortest distance between the two most distant nodes in the network. In other words, once the shortest path length from every node to all other nodes is calculated, the diameter is the longest of all the calculated path lengths. The diameter is representative of the linear size of a network. If node A-B-C-D are connected, going from A->D this would be the diameter of 3 (3-hops, 3-links).

<div class="mw-heading mw-heading3"><h3 id="Clustering_coefficient">Clustering coefficient</h3></div>

The clustering coefficient is a measure of an "all-my-friends-know-each-other" property. This is sometimes described as the friends of my friends are my friends. More precisely, the clustering coefficient of a node is the ratio of existing links connecting a node's neighbors to each other to the maximum possible number of such links. The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes. A high clustering coefficient for a network is another indication of a <a href="Small-world_experiment" title="Small-world experiment">small world</a>.

The clustering coefficient of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">'th node is

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}={2e_{i} \over k_{i}{(k_{i}-1)}}\,,}</annotation>

</semantics>

</math><img src="./3a31818a301f1761850f4812e97e9144f4153d46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.264ex; height:6.009ex;" alt="{\displaystyle C_{i}={2e_{i} \over k_{i}{(k_{i}-1)}}\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation>

</semantics>

</math><img src="./f29138ed3ad54ffce527daccadc49c520459b0b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}" loading="lazy"> is the number of neighbours of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">'th node, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

</semantics>

</math><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"> is the number of connections between these neighbours. The maximum possible number of connections between neighbors is, then,

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\binom {k}{2}}={{k(k-1)} \over 2}\,.}</annotation>

</semantics>

</math><img src="./0a92a8650ba3cc5e254014ad7fe418d2f1da5424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.836ex; height:6.343ex;" alt="{\displaystyle {\binom {k}{2}}={{k(k-1)} \over 2}\,.}" loading="lazy"></dd></dl>

From a probabilistic standpoint, the expected local clustering coefficient is the likelihood of a link existing between two arbitrary neighbors of the same node.

<div class="mw-heading mw-heading3"><h3 id="Connectedness">Connectedness</h3></div>

The way in which a network is connected plays a large part into how networks are analyzed and interpreted. Networks are classified in four different categories:

<ul><li>Clique/Complete Graph: a completely connected network, where all nodes are connected to every other node. These networks are symmetric in that all nodes have in-links and out-links from all others.</li>

<li>Giant Component: A single connected component which contains most of the nodes in the network.</li>

<li>Weakly Connected Component: A collection of nodes in which there exists a path from any node to any other, ignoring directionality of the edges.</li>

<li>Strongly Connected Component: A collection of nodes in which there exists a directed path from any node to any other.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Node_centrality">Node centrality</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Centrality" title="Centrality">Centrality</a></div>

Centrality indices produce rankings which seek to identify the most important nodes in a network model. Different centrality indices encode different contexts for the word "importance." The <a href="Betweenness_centrality" title="Betweenness centrality">betweenness centrality</a>, for example, considers a node highly important if it form bridges between many other nodes. The <a href="Centrality#Eigenvector_centrality" title="Centrality">eigenvalue centrality</a>, in contrast, considers a node highly important if many other highly important nodes link to it. Hundreds of such measures have been proposed in the literature.

Centrality indices are only accurate for identifying the most important nodes. The measures are seldom, if ever, meaningful for the remainder of network nodes.<a href="#cite_note-Lawyer2015-13">[13]</a><a href="#cite_note-Sikic2013-14">[14]</a> Also, their indications are only accurate within their assumed context for importance, and tend to "get it wrong" for other contexts.<a href="#cite_note-Borgatti2005-15">[15]</a> For example, imagine two separate communities whose only link is an edge between the most junior member of each community. Since any transfer from one community to the other must go over this link, the two junior members will have high betweenness centrality. But, since they are junior, (presumably) they have few connections to the "important" nodes in their community, meaning their eigenvalue centrality would be quite low.

<div class="mw-heading mw-heading3"><h3 id="Node_influence">Node influence</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Node_influence_metric" title="Node influence metric">Node influence metric</a></div>

Limitations to centrality measures have led to the development of more general measures.

Two examples are

the accessibility, which uses the diversity of random walks to measure how accessible the rest of the network is from a given start node,<a href="#cite_note-Travencolo2008-16">[16]</a>

and the expected force, derived from the expected value of the <a href="Force_of_infection" title="Force of infection">force of infection</a> generated by a node.<a href="#cite_note-Lawyer2015-13">[13]</a>

Both of these measures can be meaningfully computed from the structure of the network alone.

<div class="mw-heading mw-heading3"><h3 id="Community_structure">Community structure</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Community_structure" title="Community structure">Community structure</a></div>

Nodes in a network may be partitioned into groups representing communities. Depending on the context, communities may be distinct or overlapping. Typically, nodes in such communities will be strongly connected to other nodes in the same community, but weakly connected to nodes outside the community. In the absence of a <a href="Ground_truth" title="Ground truth">ground truth</a> describing the <a href="Community_structure" title="Community structure">community structure</a> of a specific network, several algorithms have been developed to infer possible community structures using either supervised of unsupervised clustering methods.

<div class="mw-heading mw-heading2"><h2 id="Network_models">Network models</h2></div>

Network models serve as a foundation to understanding interactions within empirical complex networks. Various <a href="Random_graph" title="Random graph">random graph</a> generation models produce network structures that may be used in comparison to real-world complex networks.

<div class="mw-heading mw-heading3"><h3 id="Erdős–Rényi_random_graph_model">Erdős–Rényi random graph model</h3></div>

The <a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a>, named for <a href="Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> and <a href="Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Alfréd Rényi</a>, is used for generating <a href="Random_graph" title="Random graph">random graphs</a> in which edges are set between nodes with equal probabilities. It can be used in the <a href="Probabilistic_method" title="Probabilistic method">probabilistic method</a> to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.

To generate an Erdős–Rényi model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n,p)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G(n,p)}</annotation>

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</math><img src="./ad8d6ba8bbe18701bed34c2d5106de6a56e35e08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.234ex; height:2.843ex;" alt="{\displaystyle G(n,p)}" loading="lazy"> two parameters must be specified: the total number of nodes n and the probability p that a random pair of nodes has an edge.

Because the model is generated without bias to particular nodes, the degree distribution is binomial: for a randomly chosen vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle P(\deg(v)=k)={n-1 \choose k}p^{k}(1-p)^{n-1-k}.}</annotation>

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</math><img src="./02f318dc5799b4ee47340adfa9a9c22ac3a5f16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.546ex; height:6.176ex;" alt="{\displaystyle P(\deg(v)=k)={n-1 \choose k}p^{k}(1-p)^{n-1-k}.}" loading="lazy"></dd></dl>

In this model the clustering coefficient is 0 <a href="Almost_surely" title="Almost surely">a.s</a>. The behavior of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n,p)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G(n,p)}</annotation>

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</math><img src="./ad8d6ba8bbe18701bed34c2d5106de6a56e35e08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.234ex; height:2.843ex;" alt="{\displaystyle G(n,p)}" loading="lazy"> can be broken into three regions.

Subcritical <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np<1}">

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<annotation encoding="application/x-tex">{\displaystyle np<1}</annotation>

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</math><img src="./192d3a7ecef39dfae3673a3945747441c3485d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.825ex; height:2.509ex;" alt="{\displaystyle np<1}" loading="lazy">: All components are simple and very small, the largest component has size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |C_{1}|=O(\log n)}">

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<annotation encoding="application/x-tex">{\displaystyle |C_{1}|=O(\log n)}</annotation>

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</math><img src="./184af8b58ff5e649deeb189818a299605f7cd44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.444ex; height:2.843ex;" alt="{\displaystyle |C_{1}|=O(\log n)}" loading="lazy">;

Critical <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np=1}">

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<annotation encoding="application/x-tex">{\displaystyle np=1}</annotation>

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</math><img src="./0092c8149b197ba65a3145fa4961252caa2e782c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.825ex; height:2.509ex;" alt="{\displaystyle np=1}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |C_{1}|=O(n^{\frac {2}{3}})}">

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<annotation encoding="application/x-tex">{\displaystyle |C_{1}|=O(n^{\frac {2}{3}})}</annotation>

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</math><img src="./c2413d1187f278a35d5b5a4c33a953478149408e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.821ex; height:4.009ex;" alt="{\displaystyle |C_{1}|=O(n^{\frac {2}{3}})}" loading="lazy">;

Supercritical <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np>1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle np>1}</annotation>

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</math><img src="./6821615b224311bf61ae09549d211c4e87128876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.825ex; height:2.509ex;" alt="{\displaystyle np>1}" loading="lazy">:<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |C_{1}|\approx yn}">

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<annotation encoding="application/x-tex">{\displaystyle |C_{1}|\approx yn}</annotation>

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</math><img src="./701a9278fad95645a34556a0e2fce4ded9f559ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.658ex; height:2.843ex;" alt="{\displaystyle |C_{1}|\approx yn}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y(np)}">

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<annotation encoding="application/x-tex">{\displaystyle y=y(np)}</annotation>

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</math><img src="./226af13cdd8d755ba2dcee17cbc17aac1eb0cadf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.783ex; height:2.843ex;" alt="{\displaystyle y=y(np)}" loading="lazy"> is the positive solution to the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-pny}=1-y}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{-pny}=1-y}</annotation>

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</math><img src="./c9a749b6dd0d1501d4172b34911fa1eba878241c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.481ex; height:2.843ex;" alt="{\displaystyle e^{-pny}=1-y}" loading="lazy">.

The largest connected component has high complexity. All other components are simple and small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |C_{2}|=O(\log n)}">

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<annotation encoding="application/x-tex">{\displaystyle |C_{2}|=O(\log n)}</annotation>

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</math><img src="./1eb6beea373e68bf32fd5faacfea779639ace071.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.444ex; height:2.843ex;" alt="{\displaystyle |C_{2}|=O(\log n)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Configuration_model">Configuration model</h3></div>

The configuration model takes a degree sequence<a href="#cite_note-17">[17]</a><a href="#cite_note-:0-18">[18]</a> or degree distribution<a href="#cite_note-19">[19]</a><a href="#cite_note-:1-20">[20]</a> (which subsequently is used to generate a degree sequence) as the input, and produces randomly connected graphs in all respects other than the degree sequence. This means that for a given choice of the degree sequence, the graph is chosen uniformly at random from the set of all graphs that comply with this degree sequence. The degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of a randomly chosen vertex is an <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a> random variable with integer values. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {E} [k^{2}]-2\mathbb {E} [k]>0}">

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<annotation encoding="application/x-tex">{\textstyle \mathbb {E} [k^{2}]-2\mathbb {E} [k]>0}</annotation>

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</math><img src="./0ff88c3ea2dadc4a455b28db30ca450bca41392a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.428ex; height:3.009ex;" alt="{\textstyle \mathbb {E} [k^{2}]-2\mathbb {E} [k]>0}" loading="lazy">, the configuration graph contains the <a href="Giant_component" title="Giant component">giant connected component</a>, which has infinite size.<a href="#cite_note-:0-18">[18]</a> The rest of the components have finite sizes, which can be quantified with the notion of the size distribution. The probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(n)}">

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<annotation encoding="application/x-tex">{\displaystyle w(n)}</annotation>

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</math><img src="./ccac808dddf048225ea15a41754004073012df64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.868ex; height:2.843ex;" alt="{\displaystyle w(n)}" loading="lazy"> that a randomly sampled node is connected to a component of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is given by <a href="Convolution_power" title="Convolution power">convolution powers</a> of the degree distribution:<a href="#cite_note-21">[21]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(n)={\begin{cases}{\frac {\mathbb {E} [k]}{n-1}}u_{1}^{*n}(n-2),&n>1,\\u(0)&n=1,\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle w(n)={\begin{cases}{\frac {\mathbb {E} [k]}{n-1}}u_{1}^{*n}(n-2),&n>1,\\u(0)&n=1,\end{cases}}}</annotation>

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</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(k)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle u(k)}</annotation>

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</math><img src="./2a27cb9223a4d0dac9b506e94d6b4c0848773bf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle u(k)}" loading="lazy"> denotes the degree distribution and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(k)={\frac {(k+1)u(k+1)}{\mathbb {E} [k]}}}">

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<annotation encoding="application/x-tex">{\displaystyle u_{1}(k)={\frac {(k+1)u(k+1)}{\mathbb {E} [k]}}}</annotation>

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</math><img src="./a41ff6ac7b4f61501567d622a742ed11a1d31642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.715ex; height:6.509ex;" alt="{\displaystyle u_{1}(k)={\frac {(k+1)u(k+1)}{\mathbb {E} [k]}}}" loading="lazy">. The giant component can be destroyed by randomly removing the critical fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{c}}</annotation>

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</math><img src="./836639c3805ca867b1ff24dc6db7a6b24fc69158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle p_{c}}" loading="lazy"> of all edges. This process is called <a href="Percolation_theory" title="Percolation theory">percolation on random networks</a>. When the second moment of the degree distribution is finite, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {E} [k^{2}]<\infty }">

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<annotation encoding="application/x-tex">{\textstyle \mathbb {E} [k^{2}]<\infty }</annotation>

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</math><img src="./bf57034dcb7cbf21f656c78d1d1da9ed37061373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.532ex; height:3.009ex;" alt="{\textstyle \mathbb {E} [k^{2}]<\infty }" loading="lazy">, this critical edge fraction is given by<a href="#cite_note-22">[22]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}=1-{\frac {\mathbb {E} [k]}{\mathbb {E} [k^{2}]-\mathbb {E} [k]}}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{c}=1-{\frac {\mathbb {E} [k]}{\mathbb {E} [k^{2}]-\mathbb {E} [k]}}}</annotation>

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</math><img src="./9321ee293780dff5489d984a00ab1ccf9d42d0e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:22.146ex; height:6.509ex;" alt="{\displaystyle p_{c}=1-{\frac {\mathbb {E} [k]}{\mathbb {E} [k^{2}]-\mathbb {E} [k]}}}" loading="lazy">, and the <a href="Average_path_length" title="Average path length">average vertex-vertex distance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">

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<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

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</math><img src="./829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"> in the giant component scales logarithmically with the total size of the network, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l=O(\log N)}">

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<annotation encoding="application/x-tex">{\displaystyle l=O(\log N)}</annotation>

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</math><img src="./881f3cff6fec5ead1d9e8ec56e70abbd9e64020d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.797ex; height:2.843ex;" alt="{\displaystyle l=O(\log N)}" loading="lazy">.<a href="#cite_note-:1-20">[20]</a>

In the directed configuration model, the degree of a node is given by two numbers, in-degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\text{in}}}">

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</math><img src="./bf6d24f73b7bfacfbf145cbda755bebc5687db03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.815ex; height:2.509ex;" alt="{\displaystyle k_{\text{in}}}" loading="lazy"> and out-degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\text{out}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\text{out}}}</annotation>

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</math><img src="./29d007c86497074b0e9232aef02a5cc155c7926a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.819ex; height:2.509ex;" alt="{\displaystyle k_{\text{out}}}" loading="lazy">, and consequently, the degree distribution is two-variate. The expected number of in-edges and out-edges coincides, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {E} [k_{\text{in}}]=\mathbb {E} [k_{\text{out}}]}">

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<annotation encoding="application/x-tex">{\textstyle \mathbb {E} [k_{\text{in}}]=\mathbb {E} [k_{\text{out}}]}</annotation>

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</math><img src="./7711011999f3ae1f5e9869f461b36560faa11753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.42ex; height:2.843ex;" alt="{\textstyle \mathbb {E} [k_{\text{in}}]=\mathbb {E} [k_{\text{out}}]}" loading="lazy">. The directed configuration model contains the <a href="Giant_component" title="Giant component">giant component</a> iff<a href="#cite_note-23">[23]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}k_{\text{out}}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}^{2}]+\mathbb {E} [k_{\text{in}}^{2}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}k_{\text{out}}]^{2}>0.}">

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<msub>

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<msub>

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</msub>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}k_{\text{out}}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}^{2}]+\mathbb {E} [k_{\text{in}}^{2}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}k_{\text{out}}]^{2}>0.}</annotation>

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</math>Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {E} [k_{\text{in}}]}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbb {E} [k_{\text{in}}]}</annotation>

</semantics>

</math><img src="./ff9312b027b54336305ab8ff12963de83199b91f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.659ex; height:2.843ex;" alt="{\textstyle \mathbb {E} [k_{\text{in}}]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {E} [k_{\text{out}}]}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbb {E} [k_{\text{out}}]}</annotation>

</semantics>

</math><img src="./98c4bedbb3901009ea8bcacce7b302f14eff891e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.663ex; height:2.843ex;" alt="{\textstyle \mathbb {E} [k_{\text{out}}]}" loading="lazy"> are equal and therefore interchangeable in the latter inequality. The probability that a randomly chosen vertex belongs to a component of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is given by:<a href="#cite_note-24">[24]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\text{in}}(n)={\frac {\mathbb {E} [k_{in}]}{n-1}}{\tilde {u}}_{\text{in}}^{*n}(n-2),\;n>1,\;{\tilde {u}}_{\text{in}}={\frac {k_{\text{in}}+1}{\mathbb {E} [k_{\text{in}}]}}\sum \limits _{k_{\text{out}}\geq 0}u(k_{\text{in}}+1,k_{\text{out}}),}">

<msub>

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<mfrac>

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<mrow>

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<mover>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle h_{\text{in}}(n)={\frac {\mathbb {E} [k_{in}]}{n-1}}{\tilde {u}}_{\text{in}}^{*n}(n-2),\;n>1,\;{\tilde {u}}_{\text{in}}={\frac {k_{\text{in}}+1}{\mathbb {E} [k_{\text{in}}]}}\sum \limits _{k_{\text{out}}\geq 0}u(k_{\text{in}}+1,k_{\text{out}}),}</annotation>

</semantics>

</math>for in-components, and

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\text{out}}(n)={\frac {\mathbb {E} [k_{\text{out}}]}{n-1}}{\tilde {u}}_{\text{out}}^{*n}(n-2),\;n>1,\;{\tilde {u}}_{\text{out}}={\frac {k_{\text{out}}+1}{\mathbb {E} [k_{\text{out}}]}}\sum \limits _{k_{\text{in}}\geq 0}u(k_{\text{in}},k_{\text{out}}+1),}">

<msub>

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<mfrac>

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<mover>

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<msub>

<mover>

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<mfrac>

<mrow>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{\text{out}}(n)={\frac {\mathbb {E} [k_{\text{out}}]}{n-1}}{\tilde {u}}_{\text{out}}^{*n}(n-2),\;n>1,\;{\tilde {u}}_{\text{out}}={\frac {k_{\text{out}}+1}{\mathbb {E} [k_{\text{out}}]}}\sum \limits _{k_{\text{in}}\geq 0}u(k_{\text{in}},k_{\text{out}}+1),}</annotation>

</semantics>

</math><img src="./48b3084b675547b0cb82a161fad7d2ca84de575a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:74.066ex; height:7.176ex;" alt="{\displaystyle h_{\text{out}}(n)={\frac {\mathbb {E} [k_{\text{out}}]}{n-1}}{\tilde {u}}_{\text{out}}^{*n}(n-2),\;n>1,\;{\tilde {u}}_{\text{out}}={\frac {k_{\text{out}}+1}{\mathbb {E} [k_{\text{out}}]}}\sum \limits _{k_{\text{in}}\geq 0}u(k_{\text{in}},k_{\text{out}}+1),}" loading="lazy"></dd></dl>

for out-components.

<div class="mw-heading mw-heading3"><h3 id="Watts–Strogatz_small_world_model">Watts–Strogatz small world model</h3></div>

The <a href="Watts_and_Strogatz_model" class="mw-redirect" title="Watts and Strogatz model">Watts and Strogatz model</a> is a random graph generation model that produces graphs with <a href="Small-world_properties" class="mw-redirect" title="Small-world properties">small-world properties</a>.

An initial lattice structure is used to generate a Watts–Strogatz model. Each node in the network is initially linked to its <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle k\rangle }</annotation>

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</math><img src="./79c8e81fd47c64b42f310aa18c5197183dcbb0d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.021ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle }" loading="lazy"> closest neighbors. Another parameter is specified as the rewiring probability. Each edge has a probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> that it will be rewired to the graph as a random edge. The expected number of rewired links in the model is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pE=pN\langle k\rangle /2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle pE=pN\langle k\rangle /2}</annotation>

</semantics>

</math><img src="./826ff24914918a6a0ddffc15df961a011ef99184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.711ex; height:2.843ex;" alt="{\displaystyle pE=pN\langle k\rangle /2}" loading="lazy">.

As the Watts–Strogatz model begins as a non-random lattice structure, it has a very high clustering coefficient along with a high average path length. Each rewire is likely to create a shortcut between highly connected clusters. As the rewiring probability increases, the clustering coefficient decreases slower than the average path length. In effect, this allows the average path length of the network to decrease significantly with only slight decreases in the clustering coefficient. Higher values of p force more rewired edges, which, in effect, makes the Watts–Strogatz model a random network.

<div class="mw-heading mw-heading3"><h3 id="Barabási–Albert_(BA)_preferential_attachment_model">Barabási–Albert (BA) preferential attachment model</h3></div>

The <a href="Barab%C3%A1si%E2%80%93Albert_model" title="Barabási–Albert model">Barabási–Albert model</a> is a random network model used to demonstrate a preferential attachment or a "rich-get-richer" effect. In this model, an edge is most likely to attach to nodes with higher degrees.

The network begins with an initial network of m0 nodes. m0 ≥ 2 and the degree of each node in the initial network should be at least 1, otherwise it will always remain disconnected from the rest of the network.

In the BA model, new nodes are added to the network one at a time. Each new node is connected to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability pi that the new node is connected to node i is<a href="#cite_note-RMP-25">[25]</a>

<msub>

</mrow>

</msub>

<mfrac>

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<mrow>

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</munder>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{i}={\frac {k_{i}}{\sum _{j}k_{j}}},}</annotation>

</semantics>

</math><img src="./a1bfcf374deb958fc3727dcf28c59e3647cd5278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:12.512ex; height:6.509ex;" alt="{\displaystyle p_{i}={\frac {k_{i}}{\sum _{j}k_{j}}},}" loading="lazy"></dd></dl>

where ki is the degree of node i. Heavily linked nodes ("hubs") tend to quickly accumulate even more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodes have a "preference" to attach themselves to the already heavily linked nodes.

The degree distribution resulting from the BA model is scale free, in particular, for large degree it is a power law of the form:

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle P(k)\sim k^{-3}\,}</annotation>

</semantics>

</math><img src="./eb1fd339d6c23968d5ff519fa886c465d454fd37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.796ex; height:3.176ex;" alt="{\displaystyle P(k)\sim k^{-3}\,}" loading="lazy"></dd></dl>

Hubs exhibit high betweenness centrality which allows short paths to exist between nodes. As a result, the BA model tends to have very short average path lengths. The clustering coefficient of this model also tends to 0.

The <a href="Barab%C3%A1si%E2%80%93Albert_model" title="Barabási–Albert model">Barabási–Albert model</a><a href="#cite_note-Barabasi1999-26">[26]</a> was developed for undirected networks, aiming to explain the universality of the scale-free property, and applied to a wide range of different networks and applications. The directed version of this model is the <a href="Price's_model" title="Price's model">Price model</a><a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a> which was developed to just citation networks.

<div class="mw-heading mw-heading4"><h4 id="Non-linear_preferential_attachment">Non-linear preferential attachment</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Non-linear_preferential_attachment" title="Non-linear preferential attachment">Non-linear preferential attachment</a></div>

In non-linear preferential attachment (NLPA), existing nodes in the network gain new edges proportionally to the node degree raised to a constant positive power, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">.<a href="#cite_note-29">[29]</a> Formally, this means that the probability that node <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> gains a new edge is given by

<msub>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle p_{i}={\frac {k_{i}^{\alpha }}{\sum _{j}k_{j}^{\alpha }}}.}</annotation>

</semantics>

</math><img src="./c4046e5b08b33587bc0bff20e441ee9aa8fc5e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:12.886ex; height:6.843ex;" alt="{\displaystyle p_{i}={\frac {k_{i}^{\alpha }}{\sum _{j}k_{j}^{\alpha }}}.}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =1}</annotation>

</semantics>

</math><img src="./03d67a45a44be8b8f15e99b7def2b0cf0aba1717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =1}" loading="lazy">, NLPA reduces to the BA model and is referred to as "linear". If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\alpha <1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<\alpha <1}</annotation>

</semantics>

</math><img src="./b43eb26b3586c8f17272d05089e2ce832c274dea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.009ex; height:2.176ex;" alt="{\displaystyle 0<\alpha <1}" loading="lazy">, NLPA is referred to as "sub-linear" and the degree distribution of the network tends to a <a href="Stretched_exponential_function" title="Stretched exponential function">stretched exponential distribution</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha >1}</annotation>

</semantics>

</math><img src="./17d81dbbc4786493c7b8548cc324a978d7cf5dbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >1}" loading="lazy">, NLPA is referred to as "super-linear" and a small number of nodes connect to almost all other nodes in the network. For both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha <1}</annotation>

</semantics>

</math><img src="./4769a8dab3c8a2045bc128b9000da5d661f7dab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha >1}</annotation>

</semantics>

</math><img src="./17d81dbbc4786493c7b8548cc324a978d7cf5dbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >1}" loading="lazy">, the scale-free property of the network is broken in the limit of infinite system size. However, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is only slightly larger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">, NLPA may result in <a href="Degree_distribution" title="Degree distribution">degree distributions</a> which appear to be transiently scale free.<a href="#cite_note-30">[30]</a>

<div class="mw-heading mw-heading3"><h3 id="Fitness_model">Fitness model</h3></div>

Another model where the key ingredient is the nature of the vertex has been introduced by Caldarelli et al.<a href="#cite_note-31">[31]</a> Here a link is created between two vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i,j}</annotation>

</semantics>

</math><img src="./f4cbf8bbc622154cda8208d6e339495fe16a1f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" loading="lazy"> with a probability given by a linking function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\eta _{i},\eta _{j})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle f(\eta _{i},\eta _{j})}</annotation>

</semantics>

</math><img src="./c7b2946da418927b254c555c4b5ea3ac9d76b179.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.142ex; height:3.009ex;" alt="{\displaystyle f(\eta _{i},\eta _{j})}" loading="lazy"> of the <a href="Fitness_model_(network_theory)" title="Fitness model (network theory)">fitnesses</a> of the vertices involved.

The degree of a vertex i is given by <a href="#cite_note-32">[32]</a>

<msub>

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<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle k(\eta _{i})=N\int _{0}^{\infty }f(\eta _{i},\eta _{j})\rho (\eta _{j})\,d\eta _{j}}</annotation>

</semantics>

</math><img src="./2bb8c3314f5a7b66b4822de1e5bfe16d90031a63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.138ex; height:5.843ex;" alt="{\displaystyle k(\eta _{i})=N\int _{0}^{\infty }f(\eta _{i},\eta _{j})\rho (\eta _{j})\,d\eta _{j}}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(\eta _{i})}">

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle k(\eta _{i})}</annotation>

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</math><img src="./2b94b1aff7b6f1e553f562a0d2d91c1e2520c486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.976ex; height:2.843ex;" alt="{\displaystyle k(\eta _{i})}" loading="lazy"> is an invertible and increasing function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \eta _{i}}</annotation>

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</math><img src="./b336ed51c970728280e38f5a131ac52f69833c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.955ex; height:2.176ex;" alt="{\displaystyle \eta _{i}}" loading="lazy">, then

the probability distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(k)}</annotation>

</semantics>

</math><img src="./b41614fb84549b21f2c7f2793bbd8a87a2105027.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle P(k)}" loading="lazy"> is given by

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(k)=\rho (\eta (k))\cdot \eta '(k)}</annotation>

</semantics>

</math><img src="./945bc6c6a091cafe65c419c5276e817a75fbc0d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.624ex; height:3.009ex;" alt="{\displaystyle P(k)=\rho (\eta (k))\cdot \eta '(k)}" loading="lazy"></dd></dl>

As a result, if the fitnesses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math><img src="./e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> are distributed as a power law, then also the node degree does.

Less intuitively with a fast decaying probability distribution as

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (\eta )=e^{-\eta }}</annotation>

</semantics>

</math><img src="./9db9449f35ca818da3860663b9a31398c0141be2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.7ex; height:3.009ex;" alt="{\displaystyle \rho (\eta )=e^{-\eta }}" loading="lazy"> together with a linking function of the kind

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\eta _{i},\eta _{j})=\Theta (\eta _{i}+\eta _{j}-Z)}</annotation>

</semantics>

</math><img src="./59a4f799eebf741d134fd6332683064d744d1fa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.24ex; height:3.009ex;" alt="{\displaystyle f(\eta _{i},\eta _{j})=\Theta (\eta _{i}+\eta _{j}-Z)}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> a constant and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation>

</semantics>

</math><img src="./bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }" loading="lazy"> the Heavyside function, we also obtain

scale-free networks.

Such model has been successfully applied to describe trade between nations by using GDP as fitness for the various nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i,j}</annotation>

</semantics>

</math><img src="./f4cbf8bbc622154cda8208d6e339495fe16a1f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" loading="lazy"> and a linking function of the kind

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\delta \eta _{i}\eta _{j}}{1+\delta \eta _{i}\eta _{j}}}.}</annotation>

</semantics>

</math><img src="./c7c420c7d3c50d9fb9d52ab128ac98fda503de91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.555ex; height:6.509ex;" alt="{\displaystyle {\frac {\delta \eta _{i}\eta _{j}}{1+\delta \eta _{i}\eta _{j}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Exponential_random_graph_models">Exponential random graph models</h3></div>

<a href="Exponential_family_random_graph_models" title="Exponential family random graph models">Exponential Random Graph Models (ERGMs)</a> are a family of <a href="Statistical_model" title="Statistical model">statistical models</a> for analyzing data from <a href="Social_network" title="Social network">social</a> and other networks.<a href="#cite_note-lusher2012exponential-35">[35]</a> The <a href="Exponential_family" title="Exponential family">Exponential family</a> is a broad family of models for covering many types of data, not just networks. An ERGM is a model from this family which describes networks.

We adopt the notation to represent a <a href="Random_graph" title="Random graph">random graph</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\in {\mathcal {Y}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\in {\mathcal {Y}}}</annotation>

</semantics>

</math><img src="./19c2b11fd9d821424b846eca9b02674ebafe5e77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.273ex; height:2.509ex;" alt="{\displaystyle Y\in {\mathcal {Y}}}" loading="lazy"> via a set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> nodes and a collection of <a href="Social_network" title="Social network">tie</a> variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{Y_{ij}:i=1,\dots ,n;j=1,\dots ,n\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{Y_{ij}:i=1,\dots ,n;j=1,\dots ,n\}}</annotation>

</semantics>

</math><img src="./3d289fd8ea82b6bf9560ddd9771b35cb3469bb59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.552ex; height:3.009ex;" alt="{\displaystyle \{Y_{ij}:i=1,\dots ,n;j=1,\dots ,n\}}" loading="lazy">, indexed by pairs of nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ij}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ij}</annotation>

</semantics>

</math><img src="./53fcc7b57da64979c370eb150eb5a61a625a08e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.761ex; height:2.509ex;" alt="{\displaystyle ij}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{ij}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{ij}=1}</annotation>

</semantics>

</math><img src="./45540e7c2000a6e8d92d0af68ae0ab1f1dba3638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.089ex; height:2.843ex;" alt="{\displaystyle Y_{ij}=1}" loading="lazy"> if the nodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation>

</semantics>

</math><img src="./8ef21910f980c6fca2b15bee102a7a0d861ed712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" loading="lazy"> are connected by an edge and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{ij}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{ij}=0}</annotation>

</semantics>

</math><img src="./0fac79c58c92435ceab78af89aee26d02eb46be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.089ex; height:2.843ex;" alt="{\displaystyle Y_{ij}=0}" loading="lazy"> otherwise.

The basic assumption of ERGMs is that the structure in an observed graph <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> can be explained by a given vector of <a href="Sufficient_statistic" title="Sufficient statistic">sufficient statistics</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(y)}</annotation>

</semantics>

</math><img src="./b5d72edcd5a95e13c6ff608f135d7b3742e09d2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.055ex; height:2.843ex;" alt="{\displaystyle s(y)}" loading="lazy"> which are a function of the observed network and, in some cases, nodal attributes. The probability of a graph <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in {\mathcal {Y}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in {\mathcal {Y}}}</annotation>

</semantics>

</math><img src="./2fca6d018a308729c9ef591e40b7744f1c752c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.656ex; height:2.509ex;" alt="{\displaystyle y\in {\mathcal {Y}}}" loading="lazy"> in an ERGM is defined by:

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(Y=y|\theta )={\frac {\exp(\theta ^{T}s(y))}{c(\theta )}}}</annotation>

</semantics>

</math><img src="./73711a6c75eb6f9ab193deb8ed8e1f0b5423c771.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.151ex; height:6.676ex;" alt="{\displaystyle P(Y=y|\theta )={\frac {\exp(\theta ^{T}s(y))}{c(\theta )}}}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> is a vector of model parameters associated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(y)}</annotation>

</semantics>

</math><img src="./b5d72edcd5a95e13c6ff608f135d7b3742e09d2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.055ex; height:2.843ex;" alt="{\displaystyle s(y)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\theta )=\sum _{y'\in {\mathcal {Y}}}\exp(\theta ^{T}s(y'))}">

<msup>

</msup>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(\theta )=\sum _{y'\in {\mathcal {Y}}}\exp(\theta ^{T}s(y'))}</annotation>

</semantics>

</math><img src="./5e771983a3e7c3494b1edc57c98e83ec60d6f456.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:23.6ex; height:6.176ex;" alt="{\displaystyle c(\theta )=\sum _{y'\in {\mathcal {Y}}}\exp(\theta ^{T}s(y'))}" loading="lazy"> is a normalising constant.

<div class="mw-heading mw-heading2"><h2 id="Network_analysis">Network analysis</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Social_network_analysis">Social network analysis</h3></div>

<a href="Social_network" title="Social network">Social network</a> analysis examines the structure of relationships between social entities.<a href="#cite_note-Wasserman_Faust-36">[36]</a> These entities are often persons, but may also be <a href="Group_(sociology)" class="mw-redirect" title="Group (sociology)">groups</a>, <a href="Organizations" class="mw-redirect" title="Organizations">organizations</a>, <a href="Nation_states" class="mw-redirect" title="Nation states">nation states</a>, <a href="Web_sites" class="mw-redirect" title="Web sites">web sites</a>, <a href="Scientometrics" title="Scientometrics">scholarly publications</a>.

Since the 1970s, the empirical study of networks has played a central role in social science, and many of the <a href="Mathematics" title="Mathematics">mathematical</a> and <a href="Statistics" title="Statistics">statistical</a> tools used for studying networks have been first developed in <a href="Sociology" title="Sociology">sociology</a>.<a href="#cite_note-Newman-37">[37]</a> Amongst many other applications, social network analysis has been used to understand the diffusion of <a href="Diffusion_of_innovations" title="Diffusion of innovations">innovation</a>, news and <a href="Rumor_spread_in_social_network" title="Rumor spread in social network">rumors</a>. Similarly, it has been used to examine the spread of both <a href="Epidemiology" title="Epidemiology">diseases</a> and <a href="Medical_sociology" title="Medical sociology">health-related behaviors</a>. It has also been applied to the <a href="Economic_sociology" title="Economic sociology">study of markets</a>, where it has been used to examine the role of trust in <a href="Social_exchange" class="mw-redirect" title="Social exchange">exchange relationships</a> and of social mechanisms in setting prices. Similarly, it has been used to study recruitment into <a href="Political_movement" title="Political movement">political movements</a> and social organizations. It has also been used to conceptualize scientific disagreements as well as academic prestige. More recently, network analysis (and its close cousin <a href="Traffic_analysis" title="Traffic analysis">traffic analysis</a>) has gained a significant use in military intelligence, for uncovering insurgent networks of both hierarchical and <a href="Leaderless_resistance" title="Leaderless resistance">leaderless</a> nature.<a href="#cite_note-GT-33-38">[38]</a><a href="#cite_note-39">[39]</a> In <a href="Social_network_analysis_(criminology)" class="mw-redirect" title="Social network analysis (criminology)">criminology</a>, it is being used to identify influential actors in criminal gangs, offender movements, co-offending, predict criminal activities and make policies.<a href="#cite_note-40">[40]</a>

<div class="mw-heading mw-heading3"><h3 id="Dynamic_network_analysis">Dynamic network analysis</h3></div>

<a href="Dynamic_network_analysis" title="Dynamic network analysis">Dynamic network analysis</a> examines the shifting structure of relationships among different classes of entities in complex socio-technical systems effects, and reflects social stability and changes such as the emergence of new groups, topics, and leaders.<a href="#cite_note-dynamic3-41">[41]</a><a href="#cite_note-dynamic4-42">[42]</a><a href="#cite_note-43">[43]</a> Dynamic Network Analysis focuses on meta-networks composed of multiple types of nodes (entities) and <a href="Multidimensional_network" title="Multidimensional network">multiple types of links</a>. These entities can be highly varied. Examples include people, organizations, topics, resources, tasks, events, locations, and beliefs.

Dynamic network techniques are particularly useful for assessing trends and changes in networks over time, identification of emergent leaders, and examining the co-evolution of people and ideas.

<div class="mw-heading mw-heading3"><h3 id="Biological_network_analysis">Biological network analysis</h3></div>

With the recent explosion of publicly available high throughput biological data, the analysis of molecular networks has gained significant interest. The type of analysis in this content are closely related to social network analysis, but often focusing on local patterns in the network. For example, <a href="Network_motif" title="Network motif">network motifs</a> are small subgraphs that are over-represented in the network. Activity motifs are similar over-represented patterns in the attributes of nodes and edges in the network that are over represented given the network structure. The analysis of <a href="Biological_network" title="Biological network">biological networks</a> has led to the development of <a href="Network_medicine" title="Network medicine">network medicine</a>, which looks at the effect of diseases in the <a href="Interactome" title="Interactome">interactome</a>.<a href="#cite_note-44">[44]</a>

<div class="mw-heading mw-heading3"><h3 id="Semantic_network_analysis">Semantic network analysis</h3></div>

<a href="Semantic_network" title="Semantic network">Semantic network</a> analysis is a sub-field of network analysis that focuses on the relationships between words and <a href="Concept" title="Concept">concepts</a> in a network. Words are represented as nodes and their proximity or co-occurrences in the text are represented as edges. Semantic networks are therefore graphical representations of knowledge and are commonly used in <a href="Neurolinguistics" title="Neurolinguistics">neurolinguistics</a> and <a href="Natural_language_processing" title="Natural language processing">natural language processing</a> applications. Semantic network analysis is also used as a method to analyze large texts and identify the main themes and topics (e.g., of <a href="Social_media" title="Social media">social media</a> posts), to reveal biases (e.g., in news coverage), or even to map an entire research field.<a href="#cite_note-45">[45]</a>

<div class="mw-heading mw-heading3"><h3 id="Link_analysis">Link analysis</h3></div>

Link analysis is a subset of network analysis, exploring associations between objects. An example may be examining the addresses of suspects and victims, the telephone numbers they have dialed, financial transactions they have partaken in during a given timeframe, and the familial relationships between these subjects as a part of the police investigation. Link analysis here provides the crucial relationships and associations between objects of different types that are not apparent from isolated pieces of information. Computer-assisted or fully automatic computer-based link analysis is increasingly employed by <a href="Bank" title="Bank">banks</a> and <a href="Insurance" title="Insurance">insurance</a> agencies in <a href="Fraud" title="Fraud">fraud</a> detection, by telecommunication operators in telecommunication network analysis, by medical sector in <a href="Epidemiology" title="Epidemiology">epidemiology</a> and <a href="Pharmacology" title="Pharmacology">pharmacology</a>, in law enforcement <a href="Criminal_procedure" title="Criminal procedure">investigations</a>, by <a href="Search_engine" title="Search engine">search engines</a> for <a href="Relevance" title="Relevance">relevance</a> rating (and conversely by the <a href="Search_engine_spammer" class="mw-redirect" title="Search engine spammer">spammers</a> for <a href="Spamdexing" title="Spamdexing">spamdexing</a> and by business owners for <a href="Search_engine_optimization" title="Search engine optimization">search engine optimization</a>), and everywhere else where relationships between many objects have to be analyzed.

<div class="mw-heading mw-heading3"><h3 id="Pandemic_analysis">Pandemic analysis</h3></div>

The <a href="SIR_model" class="mw-redirect" title="SIR model">SIR model</a> is one of the most well known algorithms on predicting the spread of global pandemics within an infectious population.

<div class="mw-heading mw-heading4"><h4 id="Susceptible_to_infected">Susceptible to infected</h4></div>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\beta \left({\frac {1}{N}}\right)}</annotation>

</semantics>

</math><img src="./15d79c4fc3cead07a08b9c1a2f898e5d5321008b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.638ex; height:6.176ex;" alt="{\displaystyle S=\beta \left({\frac {1}{N}}\right)}" loading="lazy"></dd></dl>

The formula above describes the "force" of infection for each susceptible unit in an infectious population, where β is equivalent to the transmission rate of said disease.

To track the change of those susceptible in an infectious population:

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta S=\beta \times S{1 \over N}\,\Delta t}</annotation>

</semantics>

</math><img src="./f56d7dacc8770cd674dab07f921828ba6583b25f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.268ex; height:5.176ex;" alt="{\displaystyle \Delta S=\beta \times S{1 \over N}\,\Delta t}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Infected_to_recovered">Infected to recovered</h4></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta I=\mu I\,\Delta t}</annotation>

</semantics>

</math><img src="./8ebeff2cbef4939fb5280d2dda067c76cbd3814f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.942ex; height:2.676ex;" alt="{\displaystyle \Delta I=\mu I\,\Delta t}" loading="lazy"></dd></dl>

Over time, the number of those infected fluctuates by: the specified rate of recovery, represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> but deducted to one over the average infectious period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over \tau }}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {1 \over \tau }}</annotation>

</semantics>

</math><img src="./a2a143010799473f68b02a6a23ba0ce430461b15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.038ex; height:5.176ex;" alt="{\displaystyle {1 \over \tau }}" loading="lazy">, the numbered of infectious individuals, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">, and the change in time, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>

</semantics>

</math><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Infectious_period">Infectious period</h4></div>

Whether a population will be overcome by a pandemic, with regards to the SIR model, is dependent on the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{0}}</annotation>

</semantics>

</math><img src="./9b8916196f182fcbaaca54f931176a4a4f5769cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}" loading="lazy"> or the "average people infected by an infected individual."

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{0}=\beta \tau ={\beta \over \mu }}</annotation>

</semantics>

</math><img src="./dd5f517ce8468bae4b8a07e9bc45c0ace647fc02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.787ex; height:5.843ex;" alt="{\displaystyle R_{0}=\beta \tau ={\beta \over \mu }}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Web_link_analysis">Web link analysis</h3></div>

Several <a href="Web_search" class="mw-redirect" title="Web search">Web search</a> <a href="Ranking" title="Ranking">ranking</a> algorithms use link-based centrality metrics, including (in order of appearance) <a href="Massimo_Marchiori" title="Massimo Marchiori">Marchiori</a>'s <a href="Hyper_Search" title="Hyper Search">Hyper Search</a>, <a href="Google" title="Google">Google</a>'s <a href="PageRank" title="PageRank">PageRank</a>, Kleinberg's <a href="HITS_algorithm" title="HITS algorithm">HITS algorithm</a>, the <a href="CheiRank" title="CheiRank">CheiRank</a> and <a href="TrustRank" title="TrustRank">TrustRank</a> algorithms. Link analysis is also conducted in information science and communication science in order to understand and extract information from the structure of collections of web pages. For example, the analysis might be of the interlinking between politicians' web sites or blogs.

<div class="mw-heading mw-heading4"><h4 id="PageRank">PageRank</h4></div>

<a href="PageRank" title="PageRank">PageRank</a> works by randomly picking "nodes" or websites and then with a certain probability, "randomly jumping" to other nodes. By randomly jumping to these other nodes, it helps PageRank completely traverse the network as some webpages exist on the periphery and would not as readily be assessed.

Each node, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">, has a PageRank as defined by the sum of pages <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> that link to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> times one over the outlinks or "out-degree" of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> times the "importance" or PageRank of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">.

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=\sum _{j\rightarrow i}{1 \over N_{j}}x_{j}^{(k)}}</annotation>

</semantics>

</math><img src="./feaed1cca193225db6e9c8381f44ba1a0583f10b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.28ex; height:6.676ex;" alt="{\displaystyle x_{i}=\sum _{j\rightarrow i}{1 \over N_{j}}x_{j}^{(k)}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading5"><h5 id="Random_jumping">Random jumping</h5></div>

As explained above, PageRank enlists random jumps in attempts to assign PageRank to every website on the internet. These random jumps find websites that might not be found during the normal search methodologies such as <a href="Breadth-first_search" title="Breadth-first search">breadth-first search</a> and <a href="Depth-first_search" title="Depth-first search">depth-first search</a>.

In an improvement over the aforementioned formula for determining PageRank includes adding these random jump components. Without the random jumps, some pages would receive a PageRank of 0 which would not be good.

The first is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">, or the probability that a random jump will occur. Contrasting is the "damping factor", or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-\alpha }</annotation>

</semantics>

</math><img src="./9afa7876fb8b4fb8c4d8039ebed6cd1cbc4781cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle 1-\alpha }" loading="lazy">.

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

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</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R{(p)}={\alpha \over N}+(1-\alpha )\sum _{j\rightarrow i}{1 \over N_{j}}x_{j}^{(k)}}</annotation>

</semantics>

</math><img src="./ecee7109b33af3049ce684200893b9204fdfb5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.32ex; height:6.676ex;" alt="{\displaystyle R{(p)}={\alpha \over N}+(1-\alpha )\sum _{j\rightarrow i}{1 \over N_{j}}x_{j}^{(k)}}" loading="lazy"></dd></dl>

Another way of looking at it:

<mfrac>

<msub>

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</msub>

<msub>

<mtext>(outlinks)</mtext>

</mrow>

</msub>

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<mfrac>

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<msub>

<mtext>(outlinks)</mtext>

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<annotation encoding="application/x-tex">{\displaystyle R(A)=\sum {R_{B} \over B_{\text{(outlinks)}}}+\cdots +{R_{n} \over n_{\text{(outlinks)}}}}</annotation>

</semantics>

</math><img src="./3c24a24cc7ce105ab16628988159b87151d63ab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.856ex; height:6.009ex;" alt="{\displaystyle R(A)=\sum {R_{B} \over B_{\text{(outlinks)}}}+\cdots +{R_{n} \over n_{\text{(outlinks)}}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Centrality_measures">Centrality measures</h3></div>

Information about the relative importance of nodes and edges in a graph can be obtained through <a href="Centrality" title="Centrality">centrality</a> measures, widely used in disciplines like <a href="Sociology" title="Sociology">sociology</a>. Centrality measures are essential when a network analysis has to answer questions such as: "Which nodes in the network should be targeted to ensure that a message or information spreads to all or most nodes in the network?" or conversely, "Which nodes should be targeted to curtail the spread of a disease?". Formally established measures of centrality are <a href="Degree_centrality" class="mw-redirect" title="Degree centrality">degree centrality</a>, <a href="Closeness_centrality" title="Closeness centrality">closeness centrality</a>, <a href="Betweenness_centrality" title="Betweenness centrality">betweenness centrality</a>, <a href="Eigenvector_centrality" title="Eigenvector centrality">eigenvector centrality</a>, and <a href="Katz_centrality" title="Katz centrality">katz centrality</a>. The objective of network analysis generally determines the type of centrality measure(s) to be used.<a href="#cite_note-Wasserman_Faust-36">[36]</a>

<ul><li>Degree centrality of a node in a network is the number of links (vertices) incident on the node.</li>

<li>Closeness centrality determines how "close" a node is to other nodes in a network by measuring the sum of the shortest distances (geodesic paths) between that node and all other nodes in the network.</li>

<li>Betweenness centrality determines the relative importance of a node by measuring the amount of traffic flowing through that node to other nodes in the network. This is done by measuring the fraction of paths connecting all pairs of nodes and containing the node of interest. Group Betweenness centrality measures the amount of traffic flowing through a group of nodes.</li>

<li>Eigenvector centrality is a more sophisticated version of degree centrality where the centrality of a node not only depends on the number of links incident on the node but also the quality of those links. This quality factor is determined by the eigenvectors of the adjacency matrix of the network.</li>

<li>Katz centrality of a node is measured by summing the geodesic paths between that node and all (reachable) nodes in the network. These paths are weighted, paths connecting the node with its immediate neighbors carry higher weights than those which connect with nodes farther away from the immediate neighbors.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Spread_of_content_in_networks">Spread of content in networks</h2></div>

Content in a <a href="Complex_network" title="Complex network">complex network</a> can spread via two major methods: conserved spread and non-conserved spread.<a href="#cite_note-46">[46]</a> In conserved spread, the total amount of content that enters a complex network remains constant as it passes through. The model of conserved spread can best be represented by a pitcher containing a fixed amount of water being poured into a series of funnels connected by tubes. The pitcher represents the source, and the water represents the spread content. The funnels and connecting tubing represent the nodes and the connections between nodes, respectively. As the water passes from one funnel into another, the water disappears instantly from the funnel that was previously exposed to the water. In non-conserved spread, the content changes as it enters and passes through a complex network. The model of non-conserved spread can best be represented by a continuously running faucet running through a series of funnels connected by tubes. Here, the amount of water from the source is infinite. Also, any funnels exposed to the water continue to experience the water even as it passes into successive funnels. The non-conserved model is the most suitable for explaining the transmission of most <a href="Infectious_diseases" class="mw-redirect" title="Infectious diseases">infectious diseases</a>.

<div class="mw-heading mw-heading3"><h3 id="The_SIR_model">The SIR model</h3></div>

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments, susceptible: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(t)}</annotation>

</semantics>

</math><img src="./ed5c98226fcae6540afa928ccb8c2245844ac0a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.148ex; height:2.843ex;" alt="{\displaystyle S(t)}" loading="lazy">, infected, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(t)}</annotation>

</semantics>

</math><img src="./e2434c9d80c34c95e25cc81ba6700f756a29dac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.821ex; height:2.843ex;" alt="{\displaystyle I(t)}" loading="lazy">, and recovered, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(t)}</annotation>

</semantics>

</math><img src="./83f5825ff7b37103bb1978b2cecddf8423a0d3f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.413ex; height:2.843ex;" alt="{\displaystyle R(t)}" loading="lazy">. The compartments used for this model consist of three classes:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(t)}</annotation>

</semantics>

</math><img src="./ed5c98226fcae6540afa928ccb8c2245844ac0a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.148ex; height:2.843ex;" alt="{\displaystyle S(t)}" loading="lazy"> is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(t)}</annotation>

</semantics>

</math><img src="./e2434c9d80c34c95e25cc81ba6700f756a29dac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.821ex; height:2.843ex;" alt="{\displaystyle I(t)}" loading="lazy"> denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(t)}</annotation>

</semantics>

</math><img src="./83f5825ff7b37103bb1978b2cecddf8423a0d3f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.413ex; height:2.843ex;" alt="{\displaystyle R(t)}" loading="lazy"> is the compartment used for those individuals who have been infected and then recovered from the disease. Those in this category are not able to be infected again or to transmit the infection to others.</li></ul>

The flow of this model may be considered as follows:

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}\rightarrow {\mathcal {I}}\rightarrow {\mathcal {R}}}</annotation>

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</math><img src="./5a4c0ea12da979998c88eea78b39d86c8928eade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.183ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}\rightarrow {\mathcal {I}}\rightarrow {\mathcal {R}}}" loading="lazy"></dd></dl>

Using a fixed population, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=S(t)+I(t)+R(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N=S(t)+I(t)+R(t)}</annotation>

</semantics>

</math><img src="./60ffc7522b0f1e1e2a00e1520c66af59b9a3f260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.225ex; height:2.843ex;" alt="{\displaystyle N=S(t)+I(t)+R(t)}" loading="lazy">, Kermack and McKendrick derived the following equations:

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=-\beta SI\\[8pt]{\frac {dI}{dt}}&=\beta SI-\gamma I\\[8pt]{\frac {dR}{dt}}&=\gamma I\end{aligned}}}</annotation>

</semantics>

</math><img src="./1572fa230e4406d5b583a221645e14b817509d4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.296ex; margin-bottom: -0.209ex; width:16.943ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {dS}{dt}}&=-\beta SI\\[8pt]{\frac {dI}{dt}}&=\beta SI-\gamma I\\[8pt]{\frac {dR}{dt}}&=\gamma I\end{aligned}}}" loading="lazy"></dd></dl>

Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy">, which is considered the contact or infection rate of the disease. Therefore, an infected individual makes contact and is able to transmit the disease with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta N}</annotation>

</semantics>

</math><img src="./a3bec44ce4637f3d5596946f51c2009c3c0b0c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.396ex; height:2.509ex;" alt="{\displaystyle \beta N}" loading="lazy"> others per unit time and the fraction of contacts by an infected with a susceptible is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S/N}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle S/N}</annotation>

</semantics>

</math><img src="./0993d8066541c7d5a6c9c6c94a8e9cc4eb5bb08e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.725ex; height:2.843ex;" alt="{\displaystyle S/N}" loading="lazy">. The number of new infections in unit time per infective then is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta N(S/N)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta N(S/N)}</annotation>

</semantics>

</math><img src="./9edd07216f1a87bdf0c33dc0c31f7128e524460d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.93ex; height:2.843ex;" alt="{\displaystyle \beta N(S/N)}" loading="lazy">, giving the rate of new infections (or those leaving the susceptible category) as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta N(S/N)I=\beta SI}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta N(S/N)I=\beta SI}</annotation>

</semantics>

</math><img src="./1cd0d4394ef1ccfaee92bb0d8ae24304600f7135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.203ex; height:2.843ex;" alt="{\displaystyle \beta N(S/N)I=\beta SI}" loading="lazy"> (Brauer & Castillo-Chavez, 2001). For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, infectives are leaving this class per unit time to enter the recovered/removed class at a rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> per unit time (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> represents the mean recovery rate, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\gamma }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/\gamma }</annotation>

</semantics>

</math><img src="./f41af61a845773b3cb991396087633e48ed5e36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.587ex; height:2.843ex;" alt="{\displaystyle 1/\gamma }" loading="lazy"> the mean infective period). These processes which occur simultaneously are referred to as the <a href="Law_of_mass_action" title="Law of mass action">Law of Mass Action</a>, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned (Daley & Gani, 2005). Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.

More can be read on this model on the <a href="Epidemic_model" class="mw-redirect" title="Epidemic model">Epidemic model</a> page.

<div class="mw-heading mw-heading3"><h3 id="The_master_equation_approach">The master equation approach</h3></div>

A <a href="Master_equation" title="Master equation">master equation</a> can express the behaviour of an undirected growing network where, at each time step, a new node is added to the network, linked to an old node (randomly chosen and without preference). The initial network is formed by two nodes and two links between them at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=2}</annotation>

</semantics>

</math><img src="./f76b791ec417942f6edf35e33b99613ca6cbaa04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=2}" loading="lazy">, this configuration is necessary only to simplify further calculations, so at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=n}</annotation>

</semantics>

</math><img src="./f08b7e09d251558e937d42ac00924c0025c3ad64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.333ex; height:2.009ex;" alt="{\displaystyle t=n}" loading="lazy"> the network have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> nodes and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> links.

The master equation for this network is:

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(k,s,t+1)={\frac {1}{t}}p(k-1,s,t)+\left(1-{\frac {1}{t}}\right)p(k,s,t),}</annotation>

</semantics>

</math><img src="./340802a3eb87f45b1384cb63fb56b9b27cea1272.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:51.053ex; height:6.176ex;" alt="{\displaystyle p(k,s,t+1)={\frac {1}{t}}p(k-1,s,t)+\left(1-{\frac {1}{t}}\right)p(k,s,t),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k,s,t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(k,s,t)}</annotation>

</semantics>

</math><img src="./24a17013f28377d03f593bd5bd45b48426074402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.277ex; height:2.843ex;" alt="{\displaystyle p(k,s,t)}" loading="lazy"> is the probability to have the node <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> with degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t+1}</annotation>

</semantics>

</math><img src="./ab2785d8415d6902b0c93efe1419c4bc3ce4643d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.842ex; height:2.343ex;" alt="{\displaystyle t+1}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> is the time step when this node was added to the network. Note that there are only two ways for an old node <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> to have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> links at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t+1}</annotation>

</semantics>

</math><img src="./ab2785d8415d6902b0c93efe1419c4bc3ce4643d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.842ex; height:2.343ex;" alt="{\displaystyle t+1}" loading="lazy">:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> have degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k-1}</annotation>

</semantics>

</math><img src="./21363ebd7038c93aae93127e7d910fc1b2e2c745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}" loading="lazy"> at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> and will be linked by the new node with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/t}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/t}</annotation>

</semantics>

</math><img src="./ac33aabb19fba3d5d9b2e6008f61658ca2a3af3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.165ex; height:2.843ex;" alt="{\displaystyle 1/t}" loading="lazy"></li>

<li>Already has degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> and will not be linked by the new node.</li></ul>

After simplifying this model, the degree distribution is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(k)=2^{-k}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(k)=2^{-k}.}</annotation>

</semantics>

</math><img src="./f3201b12631e4acd2726011900179e771c7d0d4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.041ex; height:3.176ex;" alt="{\displaystyle P(k)=2^{-k}.}" loading="lazy"><a href="#cite_note-dorogovtsev-mendes-47">[47]</a>

Based on this growing network, an epidemic model is developed following a simple rule: Each time the new node is added and after choosing the old node to link, a decision is made: whether or not this new node will be infected. The master equation for this epidemic model is:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{r}(k,s,t)=r_{t}{\frac {1}{t}}p_{r}(k-1,s,t)+\left(1-{\frac {1}{t}}\right)p_{r}(k,s,t),}</annotation>

</semantics>

</math><img src="./80dae2d69e4a4ade021e6805acb12ca2930f371f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:51.846ex; height:6.176ex;" alt="{\displaystyle p_{r}(k,s,t)=r_{t}{\frac {1}{t}}p_{r}(k-1,s,t)+\left(1-{\frac {1}{t}}\right)p_{r}(k,s,t),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{t}}</annotation>

</semantics>

</math><img src="./fb555a4a6332d0b3c8f786c87eccda2e940936d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.875ex; height:2.009ex;" alt="{\displaystyle r_{t}}" loading="lazy"> represents the decision to infect (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{t}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{t}=1}</annotation>

</semantics>

</math><img src="./606d45bafd8f0ed93751ee38843dee96114d75fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.136ex; height:2.509ex;" alt="{\displaystyle r_{t}=1}" loading="lazy">) or not (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{t}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{t}=0}</annotation>

</semantics>

</math><img src="./0246fbdd47911f6400dbc1f4213d967f12d7f4a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.136ex; height:2.509ex;" alt="{\displaystyle r_{t}=0}" loading="lazy">). Solving this master equation, the following solution is obtained: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {P}}_{r}(k)=\left({\frac {r}{2}}\right)^{k}.}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {P}}_{r}(k)=\left({\frac {r}{2}}\right)^{k}.}</annotation>

</semantics>

</math><img src="./eeea3191a012f50cc8d358b00f8878cfa9e2cbcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.414ex; height:5.343ex;" alt="{\displaystyle {\tilde {P}}_{r}(k)=\left({\frac {r}{2}}\right)^{k}.}" loading="lazy"><a href="#cite_note-cotacallapa-hase-48">[48]</a>

<div class="mw-heading mw-heading2"><h2 id="Multilayer_networks">Multilayer networks</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multidimensional_network" title="Multidimensional network">Multidimensional network</a></div>

Multilayer networks are networks with multiple kinds of relations.<a href="#cite_note-49">[49]</a> Attempts to model real-world systems as multidimensional networks have been used in various fields such as social network analysis,<a href="#cite_note-50">[50]</a> economics, history, urban and international transport, ecology, psychology, medicine, biology, commerce, climatology, physics, computational neuroscience, operations management, and finance.

<div class="mw-heading mw-heading2"><h2 id="Network_optimization">Network optimization</h2></div>

Network problems that involve finding an optimal way of doing something are studied under the name of <a href="Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a>. Examples include <a href="Flow_network" title="Flow network">network flow</a>, <a href="Shortest_path_problem" title="Shortest path problem">shortest path problem</a>, <a href="Transport_problem" class="mw-redirect" title="Transport problem">transport problem</a>, <a href="Transshipment_problem" title="Transshipment problem">transshipment problem</a>, <a href="Optimal_facility_location" title="Optimal facility location">location problem</a>, <a href="Matching_(graph_theory)" title="Matching (graph theory)">matching problem</a>, <a href="Assignment_problem" title="Assignment problem">assignment problem</a>, <a href="Packing_problem" class="mw-redirect" title="Packing problem">packing problem</a>, <a href="Routing" title="Routing">routing problem</a>, <a href="Critical_path_analysis" class="mw-redirect" title="Critical path analysis">critical path analysis</a> and <a href="PERT" class="mw-redirect" title="PERT">PERT</a> (Program Evaluation & Review Technique).

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Optimization_mechanism" title="Optimization mechanism">Optimization mechanism</a></div>

<div class="mw-heading mw-heading2"><h2 id="Interdependent_networks">Interdependent networks</h2></div>

<a href="Interdependent_networks" title="Interdependent networks">Interdependent networks</a> are networks where the functioning of nodes in one network depends on the functioning of nodes in another network. In nature, networks rarely appear in isolation, rather, usually networks are typically elements in larger systems, and interact with elements in that complex system. Such complex dependencies can have non-trivial effects on one another. A well studied example is the interdependency of infrastructure networks,<a href="#cite_note-51">[51]</a> the power stations which form the nodes of the power grid require fuel delivered via a network of roads or pipes and are also controlled via the nodes of communications network. Though the transportation network does not depend on the power network to function, the communications network does. In such infrastructure networks, the disfunction of a critical number of nodes in either the power network or the communication network can lead to cascading failures across the system with potentially catastrophic result to the whole system functioning.<a href="#cite_note-52">[52]</a> If the two networks were treated in isolation, this important feedback effect would not be seen and predictions of network robustness would be greatly overestimated.

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<ul><li><a href="Cascading_failure" title="Cascading failure">Cascading failure</a></li>

<li><a href="Climate_as_complex_networks" title="Climate as complex networks">Climate as complex networks</a></li>

<li><a href="Collaborative_innovation_network" title="Collaborative innovation network">Collaborative innovation network</a></li>

<li><a href="Communicative_ecology" title="Communicative ecology">Communicative ecology</a></li>

<li><a href="Complex_network" title="Complex network">Complex network</a></li>

<li><a href="Core-periphery_structure" class="mw-redirect" title="Core-periphery structure">Core-periphery structures</a> in networks</li>

<li><a href="Dual-phase_evolution" title="Dual-phase evolution">Dual-phase evolution</a></li>

<li><a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a></li>

<li><a href="Glossary_of_graph_theory" title="Glossary of graph theory">Glossary of graph theory</a></li>

<li><a href="Gradient_network" title="Gradient network">Gradient network</a></li>

<li><a href="Higher_category_theory" title="Higher category theory">Higher category theory</a></li>

<li><a href="Immune_network_theory" title="Immune network theory">Immune network theory</a></li>

<li><a href="Irregular_warfare" title="Irregular warfare">Irregular warfare</a></li>

<li><a href="Network_management" title="Network management">Network analyzer</a></li>

<li><a href="Network_dynamics" title="Network dynamics">Network dynamics</a></li>

<li><a href="Network_formation" title="Network formation">Network formation</a></li>

<li><a href="Network_theory_in_risk_assessment" title="Network theory in risk assessment">Network theory in risk assessment</a></li>

<li><a href="Network_topology" title="Network topology">Network topology</a></li>

<li><a href="Networks_in_labor_economics" title="Networks in labor economics">Networks in labor economics</a></li>

<li><a href="Non-linear_preferential_attachment" title="Non-linear preferential attachment">Non-linear preferential attachment</a></li>

<li><a href="Percolation" title="Percolation">Percolation</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation theory</a></li>

<li><a href="Policy_network_analysis" title="Policy network analysis">Policy network analysis</a></li>

<li><a href="Polytely" title="Polytely">Polytely</a></li>

<li><a href="Quantum_complex_network" title="Quantum complex network">Quantum complex network</a></li>

<li><a href="Random_networks" class="mw-redirect" title="Random networks">Random networks</a></li>

<li><a href="Rumor_spread_in_social_network" title="Rumor spread in social network">Rumor spread in social network</a></li>

<li><a href="Scale-free_networks" class="mw-redirect" title="Scale-free networks">Scale-free networks</a></li>

<li><a href="Sequential_dynamical_system" title="Sequential dynamical system">Sequential dynamical system</a></li>

<li><a href="Service_network" title="Service network">Service network</a></li>

<li><a href="Small-world_networks" class="mw-redirect" title="Small-world networks">Small-world networks</a></li>

<li><a href="Structural_cut-off" title="Structural cut-off">Structural cut-off</a></li>

<li><a href="Systems_theory" title="Systems theory">Systems theory</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-dorogovtsev-mendes-47"><a href="#cite_ref-dorogovtsev-mendes_47-0">^</a> <cite id="CITEREFDorogovtsevMendes2003" class="citation book cs1">Dorogovtsev, S N; Mendes, J F F (2003). Evolution of Networks: From Biological Nets to the Internet and WWW. New York, NY, USA: Oxford University Press, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0198515906</bdi>.</cite>

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<li id="cite_note-cotacallapa-hase-48"><a href="#cite_ref-cotacallapa-hase_48-0">^</a> <cite id="CITEREFCotacallapaHase2016" class="citation journal cs1">Cotacallapa, M; Hase, M O (2016). "Epidemics in networks: a master equation approach". Journal of Physics A. 49 (6): 065001. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1604.01049">1604.01049</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016JPhA...49f5001C">2016JPhA...49f5001C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1751-8113%2F49%2F6%2F065001">10.1088/1751-8113/49/6/065001</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119206200">119206200</a>.</cite>

</li>

<li id="cite_note-49"><a href="#cite_ref-49">^</a> <cite id="CITEREFDe_Domenico2022" class="citation book cs1">De Domenico, Manlio (March 31, 2022). Multilayer Networks: Analysis and Visualization (1st ed.). Springer.</cite>

</li>

<li id="cite_note-50"><a href="#cite_ref-50">^</a> <cite id="CITEREFRossiDickisonMagnani2016" class="citation book cs1">Rossi, Luca; Dickison, Mark E.; Magnani, Matteo (July 18, 2016). Multilayer Social Networks (1st ed.). Cambridge University Press.</cite>

</li>

<li id="cite_note-51"><a href="#cite_ref-51">^</a> <cite class="citation journal cs1">"Identifying, understanding, and analyzing critical infrastructure interdependencies". IEEE Control Systems Magazine. 21 (6): 11–25. December 2001. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F37.969131">10.1109/37.969131</a>.</cite>

</li>

<li id="cite_note-52"><a href="#cite_ref-52">^</a> <cite id="CITEREFBuldyrev2010" class="citation journal cs1">Buldyrev, Sergey V.; et al. (April 2010). "Catastrophic cascade of failures in interdependent networks". Nature. 464 (7291): 1025–1028. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.1182">0907.1182</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010Natur.464.1025B">2010Natur.464.1025B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature08932">10.1038/nature08932</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20393559">20393559</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1836955">1836955</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.cambridge.org/us/academic/subjects/physics/statistical-physics/first-course-network-science">A First Course in Network Science</a>, <a href="Filippo_Menczer" title="Filippo Menczer">F. Menczer</a>, S. Fortunato, C.A. Davis. (Cambridge University Press, 2020). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781108471138</bdi>. <a rel="nofollow" class="external text" href="https://github.com/CambridgeUniversityPress/FirstCourseNetworkScience">GitHub site</a> with tutorials, datasets, and other resources</li>

<li>"Connected: The Power of Six Degrees," <a rel="nofollow" class="external free" href="https://web.archive.org/web/20111006191031/http://ivl.slis.indiana.edu/km/movies/2008-talas-connected.mov">https://web.archive.org/web/20111006191031/http://ivl.slis.indiana.edu/km/movies/2008-talas-connected.mov</a></li>

<li><cite id="CITEREFCohenErez2000" class="citation journal cs1">Cohen, R.; Erez, K. (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130512011815/http://havlin.biu.ac.il/Publications.php?keyword=Resilience+of+the+Internet+to+random+breakdown&year=*&match=all">"Resilience of the Internet to random breakdown"</a>. Phys. Rev. Lett. 85 (21): 4626–4628. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0007048">cond-mat/0007048</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000PhRvL..85.4626C">2000PhRvL..85.4626C</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.242.6797">10.1.1.242.6797</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.85.4626">10.1103/physrevlett.85.4626</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11082612">11082612</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15372152">15372152</a>. Archived from <a rel="nofollow" class="external text" href="http://havlin.biu.ac.il/Publications.php?keyword=Resilience+of+the+Internet+to+random+breakdown&year=*&match=all">the original</a> on 2013-05-12. Retrieved 2011-04-12.</cite></li>

<li><cite id="CITEREFPuWen-PeiMichaelson2012" class="citation journal cs1">Pu, Cun-Lai; Wen-; Pei, Jiang; Michaelson, Andrew (2012). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161013130539/http://www.barabasilab.com/pubs/CCNR-ALB_Publications/201204-24_PhysicaA-RobustnessControllability/201204-24_PhysicaA-RobustnessControllability.pdf">"Robustness analysis of network controllability"</a> (PDF). Physica A. 391 (18): 4420–4425. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PhyA..391.4420P">2012PhyA..391.4420P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physa.2012.04.019">10.1016/j.physa.2012.04.019</a>. Archived from <a rel="nofollow" class="external text" href="http://www.barabasilab.com/pubs/CCNR-ALB_Publications/201204-24_PhysicaA-RobustnessControllability/201204-24_PhysicaA-RobustnessControllability.pdf">the original</a> (PDF) on 2016-10-13. Retrieved 2013-09-18.</cite></li>

<li>S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Networks: From biological networks to the Internet and WWW, Oxford University Press, 2003, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-851590-1</bdi></li>

<li>Linked: The New Science of Networks, A.-L. Barabási (Perseus Publishing, Cambridge)</li>

<li>'<a rel="nofollow" class="external text" href="https://global.oup.com/academic/product/scale-free-networks-9780199665174?q=Caldarelli&lang=en&cc=it">Scale-Free Networks</a>, G. Caldarelli (Oxford University Press, Oxford)</li>

<li><a rel="nofollow" class="external text" href="http://www.nap.edu/catalog.php?record_id=11516">Network Science</a>, Committee on Network Science for Future Army Applications, National Research Council. 2005. The National Academies Press (2005)<a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-309-10026-7</bdi></li>

<li>Network Science Bulletin, USMA (2007) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-934808-00-9</bdi></li>

<li>The Structure and Dynamics of Networks Mark Newman, Albert-László Barabási, & Duncan J. Watts (The Princeton Press, 2006) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-11357-2</bdi></li>

<li>Dynamical processes on complex networks, Alain Barrat, Marc Barthelemy, Alessandro Vespignani (Cambridge University Press, 2008) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-87950-7</bdi></li>

<li>Network Science: Theory and Applications, Ted G. Lewis (Wiley, March 11, 2009) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-470-33188-7</bdi></li>

<li>Nexus: Small Worlds and the Groundbreaking Theory of Networks, Mark Buchanan (W. W. Norton & Company, June 2003) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-32442-7</bdi></li>

<li>Six Degrees: The Science of a Connected Age, Duncan J. Watts (W. W. Norton & Company, February 17, 2004) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-393-32542-3</bdi></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Social_networks_and_social_media172" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Social_networks_and_social_media172" style="font-size:114%;margin:0 4em"><a href="Social_network" title="Social network">Social networks</a> and <a href="Social_media" title="Social media">social media</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Personal_network" title="Personal network">Personal</a></li>

<li><a href="Professional_network_service" title="Professional network service">Professional</a></li>

<li><a href="Sexual_network" title="Sexual network">Sexual</a></li>

<li><a href="Value_network" title="Value network">Value</a></li>

<li><a href="Clique" title="Clique">Clique</a>

<ul><li><a href="Adolescent_clique" title="Adolescent clique">Adolescent</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Networks</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em">

<ul><li><a href="Corporate_social_media" title="Corporate social media">Corporate social media</a></li>

<li><a href="Distributed_social_network" title="Distributed social network">Distributed social network</a> (<a href="Comparison_of_software_and_protocols_for_distributed_social_networking" title="Comparison of software and protocols for distributed social networking">list</a>)</li>

<li><a href="Enterprise_social_networking" title="Enterprise social networking">Enterprise social networking</a></li>

<li><a href="Enterprise_social_software" title="Enterprise social software">Enterprise social software</a></li>

<li><a href="Mobile_social_network" title="Mobile social network">Mobile social network</a></li>

<li><a href="Personal_knowledge_networking" title="Personal knowledge networking">Personal knowledge networking</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Social_networking_service" title="Social networking service">Services</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_social_networking_services" title="List of social networking services">List of social networking services</a></li>

<li><a href="List_of_virtual_communities_with_more_than_1_million_users" title="List of virtual communities with more than 1 million users">List of virtual communities with more than 1 million users</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts and theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em">

<ul><li><a href="Ambient_awareness" title="Ambient awareness">Ambient awareness</a></li>

<li><a href="Assortative_mixing" title="Assortative mixing">Assortative mixing</a></li>

<li><a href="Attention_inequality" title="Attention inequality">Attention inequality</a></li>

<li><a href="Bridge_(interpersonal)" title="Bridge (interpersonal)">Interpersonal bridge</a></li>

<li><a href="Organizational_network_analysis" title="Organizational network analysis">Organizational network analysis</a></li>

<li><a href="Small-world_experiment" title="Small-world experiment">Small-world experiment</a></li>

<li><a href="Social_aspects_of_television" title="Social aspects of television">Social aspects of television</a></li>

<li><a href="Social_capital" title="Social capital">Social capital</a></li>

<li><a href="Social_data_revolution" title="Social data revolution">Social data revolution</a></li>

<li><a href="Social_exchange_theory" title="Social exchange theory">Social exchange theory</a></li>

<li><a href="Social_identity_theory" title="Social identity theory">Social identity theory</a></li>

<li><a href="Social_media_and_psychology" title="Social media and psychology">Social media and psychology</a></li>

<li><a href="Social_media_intelligence" title="Social media intelligence">Social media intelligence</a></li>

<li><a href="Social_media_mining" title="Social media mining">Social media mining</a></li>

<li><a href="Social_media_optimization" title="Social media optimization">Social media optimization</a></li>

<li><a href="Social_network_analysis" title="Social network analysis">Social network analysis</a></li>

<li><a href="Social_web" title="Social web">Social web</a></li>

<li><a href="Structural_endogamy" title="Structural endogamy">Structural endogamy</a></li>

<li><a href="Virtual_collective_consciousness" title="Virtual collective consciousness">Virtual collective consciousness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models and processes</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Account_verification" title="Account verification">Account verification</a></li>

<li><a href="Social_network_aggregation" title="Social network aggregation">Aggregation</a></li>

<li><a href="Social_network_change_detection" class="mw-redirect" title="Social network change detection">Change detection</a></li>

<li><a href="Blockmodeling" title="Blockmodeling">Blockmodeling</a></li>

<li><a href="Collaboration_graph" title="Collaboration graph">Collaboration graph</a></li>

<li><a href="Collaborative_consumption" title="Collaborative consumption">Collaborative consumption</a></li>

<li><a href="Giant_Global_Graph" title="Giant Global Graph">Giant Global Graph</a></li>

<li><a href="Lateral_communication" title="Lateral communication">Lateral communication</a></li>

<li><a href="Reputation_system" title="Reputation system">Reputation system</a></li>

<li><a href="Social_bot" title="Social bot">Social bot</a></li>

<li><a href="Social_graph" title="Social graph">Social graph</a></li>

<li><a href="Social_media_analytics" title="Social media analytics">Social media analytics</a></li>

<li><a href="Social_network_analysis_software" title="Social network analysis software">Social network analysis software</a></li>

<li><a href="Social_networking_potential" class="mw-redirect" title="Social networking potential">Social networking potential</a></li>

<li><a href="Social_television" title="Social television">Social television</a></li>

<li><a href="Structural_cohesion" title="Structural cohesion">Structural cohesion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Economics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em">

<ul><li><a href="Affinity_fraud" title="Affinity fraud">Affinity fraud</a></li>

<li><a href="Attention_economy" title="Attention economy">Attention economy</a></li>

<li><a href="Collaborative_finance" title="Collaborative finance">Collaborative finance</a></li>

<li><a href="Creator_economy" title="Creator economy">Creator economy</a></li>

<li><a href="Influencer_marketing" title="Influencer marketing">Influencer marketing</a></li>

<li><a href="Narrowcasting" title="Narrowcasting">Narrowcasting</a></li>

<li><a href="Sharing_economy" title="Sharing economy">Sharing economy</a></li>

<li><a href="Social_commerce" title="Social commerce">Social commerce</a></li>

<li><a href="Social_sorting" title="Social sorting">Social sorting</a></li>

<li><a href="Viral_marketing" title="Viral marketing">Viral marketing</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algorithmic_radicalization" title="Algorithmic radicalization">Algorithmic radicalization</a></li>

<li><a href="Community_recognition" title="Community recognition">Community recognition</a></li>

<li><a href="Complex_contagion" title="Complex contagion">Complex contagion</a></li>

<li><a href="Computer_addiction" title="Computer addiction">Computer addiction</a></li>

<li><a href="Consequential_strangers" title="Consequential strangers">Consequential strangers</a></li>

<li><a href="Friend_of_a_friend" title="Friend of a friend">Friend of a friend</a></li>

<li><a href="Friending_and_following" title="Friending and following">Friending and following</a></li>

<li><a href="Friendship_paradox" title="Friendship paradox">Friendship paradox</a></li>

<li><a href="Influence-for-hire" title="Influence-for-hire">Influence-for-hire</a></li>

<li><a href="Internet_addiction" class="mw-redirect" title="Internet addiction">Internet addiction</a></li>

<li><a href="Information_overload" title="Information overload">Information overload</a></li>

<li><a href="Overchoice" title="Overchoice">Overchoice</a></li>

<li><a href="Six_degrees_of_separation" title="Six degrees of separation">Six degrees of separation</a></li>

<li><a href="Social_media_addiction" class="mw-redirect" title="Social media addiction">Social media addiction</a></li>

<li><a href="Social_media_and_suicide" title="Social media and suicide">Social media and suicide</a></li>

<li><a href="Social_invisibility" title="Social invisibility">Social invisibility</a></li>

<li><a href="Social_network_game" title="Social network game">Social network game</a></li>

<li><a href="Suicide_and_the_Internet" title="Suicide and the Internet">Suicide and the Internet</a></li>

<li><a href="Tribe_(internet)" title="Tribe (internet)">Tribe</a></li>

<li><a href="Viral_phenomenon" title="Viral phenomenon">Viral phenomenon</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0;background:#F4F0E;"><div style="padding:0 0.25em">

<ul><li><a href="Friendship_recession" title="Friendship recession">Friendship recession</a></li>

<li><a href="Peer_pressure" title="Peer pressure">Peer pressure</a></li>

<li><a href="User_profile" title="User profile">User profile</a>

<ul><li><a href="Online_identity" title="Online identity">Online identity</a></li>

<li><a href="Persona_(user_experience)" title="Persona (user experience)">Persona</a></li>

<li><a href="Social_profiling" title="Social profiling">Social profiling</a></li></ul></li>

<li><a href="List_of_social_network_researchers" class="mw-redirect" title="List of social network researchers">Researchers</a></li>

<li><a href="Viral_messages" class="mw-redirect" title="Viral messages">Viral messages</a></li>

<li><a href="Virtual_community" title="Virtual community">Virtual community</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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