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<title>Random graph</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Random graph</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For the countably-infinite random graph, see <a href="Rado_graph" title="Rado graph">Rado graph</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%;"><a href="Network_science" title="Network science">Network science</a></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><span class="nowrap"><a href="Adjacency_list" title="Adjacency list">Adjacency list</a> / <a href="Adjacency_matrix" title="Adjacency matrix">matrix</a></span></li>

<li><span class="nowrap"><a href="Incidence_list" class="mw-redirect" title="Incidence list">Incidence list</a> / <a href="Incidence_matrix" title="Incidence matrix">matrix</a></span></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="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="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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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>random graph</b> is the general term to refer to <a href="Probability_distribution" title="Probability distribution">probability distributions</a> over <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>. Random graphs may be described simply by a probability distribution, or by a <a href="Random_process" class="mw-redirect" title="Random process">random process</a> which generates them.<sup id="cite_ref-Random_Graphs_1-0" class="reference"><a href="#cite_note-Random_Graphs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Introduction_to_Random_graphs_2-0" class="reference"><a href="#cite_note-Introduction_to_Random_graphs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The theory of random graphs lies at the intersection between <a href="Graph_theory" title="Graph theory">graph theory</a> and <a href="Probability_theory" title="Probability theory">probability theory</a>. From a mathematical perspective, random graphs are used to answer questions about the properties of <i>typical</i> graphs. Its practical applications are found in all areas in which <a href="Complex_network" title="Complex network">complex networks</a> need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, <i>random graph</i> refers almost exclusively to the <a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi random graph model</a>. In other contexts, any graph model may be referred to as a <i>random graph</i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Models">Models</h2></div>

<p>A random graph is obtained by starting with a set of <i>n</i> isolated vertices and adding successive edges between them at random. The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise.<sup id="cite_ref-Random_Graphs2_3-0" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Different <b>random graph models</b> produce different <a href="Probability_distribution" title="Probability distribution">probability distributions</a> on graphs. Most commonly studied is the one proposed by <a href="Edgar_Gilbert" title="Edgar Gilbert">Edgar Gilbert</a> but often called the <a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a>, denoted <i>G</i>(<i>n</i>,<i>p</i>). In it, every possible edge occurs independently with probability 0 < <i>p</i> < 1. The probability of obtaining <i>any one particular</i> random graph with <i>m</i> edges is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{m}(1-p)^{N-m}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{m}(1-p)^{N-m}}</annotation>

</semantics>

</math></span><img src="./a651740c801d762a3f4306fedc1d7acf5ab2b46b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.328ex; height:3.176ex;" alt="{\displaystyle p^{m}(1-p)^{N-m}}" loading="lazy"></span> with the notation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\tbinom {n}{2}}}">

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c60f7fe95d23ef0f6aead7ae0ee99176bcab150f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.278ex; height:3.176ex;" alt="{\displaystyle N={\tbinom {n}{2}}}" loading="lazy"></span>.<sup id="cite_ref-Random_Graphs3_4-0" class="reference"><a href="#cite_note-Random_Graphs3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>A closely related model, also called the Erdős–Rényi model and denoted <i>G</i>(<i>n</i>,<i>M</i>), assigns equal probability to all graphs with exactly <i>M</i> edges. With 0 ≤ <i>M</i> ≤ <i>N</i>, <i>G</i>(<i>n</i>,<i>M</i>) has <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {N}{M}}}">

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./dfe7c633b96c7ececeabdc5411e8e063fb61afa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.857ex; height:3.343ex;" alt="{\displaystyle {\tbinom {N}{M}}}" loading="lazy"></span> elements and every element occurs with probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\tbinom {N}{M}}}">

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9db537a91ea66a3b8e150ced34e4aa07d96d4898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.182ex; height:3.343ex;" alt="{\displaystyle 1/{\tbinom {N}{M}}}" loading="lazy"></span>.<sup id="cite_ref-Random_Graphs2_3-1" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The <i>G</i>(<i>n</i>,<i>M</i>) model can be viewed as a snapshot at a particular time (<i>M</i>) of the <b>random graph process</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {G}}_{n}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8340a778e1a962157f13ea507aff67ef7ad4e59f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.045ex; height:3.009ex;" alt="{\displaystyle {\tilde {G}}_{n}}" loading="lazy"></span>, a <a href="Stochastic_process" title="Stochastic process">stochastic process</a> that starts with <i>n</i> vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.

</p><p>If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < <i>p</i> < 1, then we get an object <i>G</i> called an <b>infinite random graph</b>. Except in the trivial cases when <i>p</i> is 0 or 1, such a <i>G</i> <a href="Almost_surely" title="Almost surely">almost surely</a> has the following property:

</p>

<blockquote><p>Given any <i>n</i> + <i>m</i> elements <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V}</annotation>

</semantics>

</math></span><img src="./91edaec20ab1447185fd3594a2d29992ca51bf22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.475ex; height:2.509ex;" alt="{\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V}" loading="lazy"></span>, there is a vertex <i>c</i> in <i>V</i> that is adjacent to each of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./451345cc97e2ed923dd4656fcc400c3f37119cca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\ldots ,a_{n}}" loading="lazy"></span> and is not adjacent to any of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1},\ldots ,b_{m}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./cd49a4e6f0dbeba4abaaae3b7987a9eec9eeac0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.903ex; height:2.509ex;" alt="{\displaystyle b_{1},\ldots ,b_{m}}" loading="lazy"></span>.</p></blockquote>

<p>It turns out that if the vertex set is <a href="Countable" class="mw-redirect" title="Countable">countable</a> then there is, <a href="Up_to" title="Up to">up to</a> <a href="Graph_isomorphism" title="Graph isomorphism">isomorphism</a>, only a single graph with this property, namely the <a href="Rado_graph" title="Rado graph">Rado graph</a>. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the <b>random graph</b>. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.

</p><p>Another model, which generalizes Gilbert's random graph model, is the <b>random dot-product model</b>. A random dot-product graph associates with each vertex a <a href="Real_vector" class="mw-redirect" title="Real vector">real vector</a>. The probability of an edge <i>uv</i> between any vertices <i>u</i> and <i>v</i> is some function of the <a href="Dot_product" title="Dot product">dot product</a> <b>u</b> • <b>v</b> of their respective vectors.

</p><p>The <a href="Network_probability_matrix" title="Network probability matrix">network probability matrix</a> models random graphs through edge probabilities, which represent the probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e82f4cb43c1f5cd53898ed7cc80dcf092373f676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:3.193ex; height:2.343ex;" alt="{\displaystyle p_{i,j}}" loading="lazy"></span> that a given edge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d9c7a1c781b037551cd83fc9f1939d47f8a3cf4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.018ex; height:2.343ex;" alt="{\displaystyle e_{i,j}}" loading="lazy"></span> exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure.

</p><p>For <i>M</i> ≃ <i>pN</i>, where <i>N</i> is the maximal number of edges possible, the two most widely used models, <i>G</i>(<i>n</i>,<i>M</i>) and <i>G</i>(<i>n</i>,<i>p</i>), are almost interchangeable.<sup id="cite_ref-Handbook_5-0" class="reference"><a href="#cite_note-Handbook-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Random_regular_graph" title="Random regular graph">Random regular graphs</a> form a special case, with properties that may differ from random graphs in general.

</p><p>Once we have a model of random graphs, every function on graphs, becomes a <a href="Random_variable" title="Random variable">random variable</a>. The study of this model is to determine if, or at least estimate the probability that, a property may occur.<sup id="cite_ref-Random_Graphs3_4-1" class="reference"><a href="#cite_note-Random_Graphs3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div>

<p>The term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the <i>error probabilities</i> tend to zero.<sup id="cite_ref-Random_Graphs3_4-2" class="reference"><a href="#cite_note-Random_Graphs3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution. For example, we might ask for a given value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> what the probability is that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span> is <a href="Connection_(mathematics)" title="Connection (mathematics)">connected</a>. In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> grows very large. <a href="Percolation_theory" title="Percolation theory">Percolation theory</a> characterizes the connectedness of random graphs, especially infinitely large ones.

</p><p>Percolation is related to the robustness of the graph (called also network). Given a random graph of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> nodes and an average degree <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>. Next we remove randomly a fraction <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9633a8692121eedfa99cace406205e5d1511ef8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}" loading="lazy"></span> of nodes and leave only a fraction <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>. There exists a critical percolation threshold <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}</annotation>

</semantics>

</math></span><img src="./a99569c6d70801594712783ce52840fb48af5195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.273ex; height:4.176ex;" alt="{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}" loading="lazy"></span> below which the network becomes fragmented while above <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> a giant connected component exists.<sup id="cite_ref-Random_Graphs_1-1" class="reference"><a href="#cite_note-Random_Graphs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Handbook_5-1" class="reference"><a href="#cite_note-Handbook-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Random_graphs_6-0" class="reference"><a href="#cite_note-Random_graphs-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-On_Random_Graphs_8-0" class="reference"><a href="#cite_note-On_Random_Graphs-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p><p>Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9633a8692121eedfa99cace406205e5d1511ef8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}" loading="lazy"></span> of nodes from the network is removed. It was shown that for random graph with Poisson distribution of degrees <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}</annotation>

</semantics>

</p><p>Random graphs are widely used in the <a href="Probabilistic_method" title="Probabilistic method">probabilistic method</a>, where one tries to prove the existence of graphs with certain properties. The existence of a property on a random graph can often imply, via the <a href="Szemer%C3%A9di_regularity_lemma" title="Szemerédi regularity lemma">Szemerédi regularity lemma</a>, the existence of that property on almost all graphs.

</p><p>In <a href="Random_regular_graph" title="Random regular graph">random regular graphs</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n,r-reg)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f56909858c0fc1449a150cc2ac7ae18460c840d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.202ex; height:2.843ex;" alt="{\displaystyle G(n,r-reg)}" loading="lazy"></span> are the set of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span>-regular graphs with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r(n)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ca5b4b5d48c9035c75618947d9051c2935c9540c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.4ex; height:2.843ex;" alt="{\displaystyle r=r(n)}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> are the natural numbers, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\leq r<n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3\leq r<n}</annotation>

</semantics>

</math></span><img src="./0d1fe2fc0a252f04bccb9a1fbe5b1d83bafaa87e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.803ex; height:2.343ex;" alt="{\displaystyle 3\leq r<n}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rn=2m}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./52459f4b0df9a3d09de83b51f46159673380e41e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.745ex; height:2.176ex;" alt="{\displaystyle rn=2m}" loading="lazy"></span> is even.<sup id="cite_ref-Random_Graphs2_3-2" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>The degree sequence of a graph <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f43241538481dd8112f3c0ada4237a2c7b2aa6e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.045ex; height:2.343ex;" alt="{\displaystyle G^{n}}" loading="lazy"></span> depends only on the number of edges in the sets<sup id="cite_ref-Random_Graphs2_3-3" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{n}^{(2)}=\left\{ij\ :\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i=1,\cdots ,n.}</annotation>

</semantics>

</math></span><img src="./b0841ba1939842ed3ce26c0f6248e9372ef08b44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.449ex; height:3.509ex;" alt="{\displaystyle V_{n}^{(2)}=\left\{ij\ :\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i=1,\cdots ,n.}" loading="lazy"></span></dd></dl>

<p>If edges, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> in a random graph, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8ab6ce334c0de8fb3417356ce6ed775812acea73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}" loading="lazy"></span> is large enough to ensure that almost every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8ab6ce334c0de8fb3417356ce6ed775812acea73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}" loading="lazy"></span> has minimum degree at least 1, then almost every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8ab6ce334c0de8fb3417356ce6ed775812acea73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}" loading="lazy"></span> is connected and, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is even, almost every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8ab6ce334c0de8fb3417356ce6ed775812acea73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}" loading="lazy"></span> has a perfect matching. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.<sup id="cite_ref-Random_Graphs2_3-4" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Almost every graph process on an even number of vertices with the edge raising the minimum degree to 1 or a random graph with slightly more than <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{4}}\log(n)}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{4}}\log(n)}</annotation>

</semantics>

</math></span><img src="./c2f7a78274dd7143981614cbdef0dfffb039321c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.385ex; height:3.176ex;" alt="{\displaystyle {\tfrac {n}{4}}\log(n)}" loading="lazy"></span> edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex.

</p><p>For some constant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>, almost every labeled graph with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> vertices and at least <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cn\log(n)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a9f3b907e2ab170f8551c908f7a4357d1d5f299f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.964ex; height:2.843ex;" alt="{\displaystyle cn\log(n)}" loading="lazy"></span> edges is <a href="Hamiltonian_cycle" class="mw-redirect" title="Hamiltonian cycle">Hamiltonian</a>. With the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian.

</p><p>Properties of random graph may change or remain invariant under graph transformations. <a href="Alireza_Mashaghi" title="Alireza Mashaghi">Mashaghi A.</a> et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Colouring">Colouring</h2></div>

<p>Given a random graph <i>G</i> of order <i>n</i> with the vertex <i>V</i>(<i>G</i>) = {1, ..., <i>n</i>}, by the <a href="Greedy_algorithm" title="Greedy algorithm">greedy algorithm</a> on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).<sup id="cite_ref-Random_Graphs2_3-5" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

The number of proper colorings of random graphs given a number of <i>q</i> colors, called its <a href="Chromatic_polynomial" title="Chromatic polynomial">chromatic polynomial</a>, remains unknown so far. The scaling of zeros of the chromatic polynomial of random graphs with parameters <i>n</i> and the number of edges <i>m</i> or the connection probability <i>p</i> has been studied empirically using an algorithm based on symbolic pattern matching.<sup id="cite_ref-Chromatic_Polynomials_of_Random_Graphs_10-0" class="reference"><a href="#cite_note-Chromatic_Polynomials_of_Random_Graphs-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Random_trees">Random trees</h2></div>

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

<p>A <a href="Random_tree" title="Random tree">random tree</a> is a <a href="Tree_(graph_theory)" title="Tree (graph theory)">tree</a> or <a href="Arborescence_(graph_theory)" title="Arborescence (graph theory)">arborescence</a> that is formed by a <a href="Stochastic_process" title="Stochastic process">stochastic process</a>. In a large range of random graphs of order <i>n</i> and size <i>M</i>(<i>n</i>) the distribution of the number of tree components of order <i>k</i> is asymptotically <a href="Poisson_distribution" title="Poisson distribution">Poisson</a>. Types of random trees include <a href="Uniform_spanning_tree" class="mw-redirect" title="Uniform spanning tree">uniform spanning tree</a>, <a href="Random_minimum_spanning_tree" title="Random minimum spanning tree">random minimum spanning tree</a>, <a href="Random_binary_tree" title="Random binary tree">random binary tree</a>, <a href="Treap" title="Treap">treap</a>, <a href="Rapidly_exploring_random_tree" title="Rapidly exploring random tree">rapidly exploring random tree</a>, <a href="Brownian_tree" title="Brownian tree">Brownian tree</a>, and <a href="Random_forest" title="Random forest">random forest</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Conditional_random_graphs">Conditional random graphs</h2></div>

<p>Consider a given random graph model defined on the probability space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

</semantics>

</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> and let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(G):\Omega \rightarrow R^{m}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(G):\Omega \rightarrow R^{m}}</annotation>

</semantics>

</math></span><img src="./a0deefd1e904627b10cae8d92d98217e5fdd7f3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.008ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(G):\Omega \rightarrow R^{m}}" loading="lazy"></span> be a real valued function which assigns to each graph in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> a vector of <i>m</i> properties.

For a fixed <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} \in R^{m}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} \in R^{m}}</annotation>

</semantics>

</math></span><img src="./d28b061fe99de21035280469087e0b2a85f3481f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.765ex; height:2.676ex;" alt="{\displaystyle \mathbf {p} \in R^{m}}" loading="lazy"></span>, <i>conditional random graphs</i> are models in which the probability measure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(G)\neq \mathbf {p} }</annotation>

</semantics>

</math></span><img src="./587c77eeb66200b4a32eefe1b599093756886d0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.923ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(G)\neq \mathbf {p} }" loading="lazy"></span>.

</p><p>Special cases are <i>conditionally uniform random graphs</i>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> assigns equal probability to all the graphs having specified properties. They can be seen as a generalization of the <a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a> <i>G</i>(<i>n</i>,<i>M</i>), when the conditioning information is not necessarily the number of edges <i>M</i>, but whatever other arbitrary graph property <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(G)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./79964cec08dc040872b71f5aa356cafb266ca5af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(G)}" loading="lazy"></span>. In this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>The earliest use of a random graph model was by <a href="Helen_Hall_Jennings" title="Helen Hall Jennings">Helen Hall Jennings</a> and <a href="Jacob_Moreno" class="mw-redirect" title="Jacob Moreno">Jacob Moreno</a> in 1938 where a "chance sociogram" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Another use, under the name "random net", was by <a href="Ray_Solomonoff" title="Ray Solomonoff">Ray Solomonoff</a> and <a href="Anatol_Rapoport" title="Anatol Rapoport">Anatol Rapoport</a> in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p><p>The <a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a> of random graphs was first defined by <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> in their 1959 paper "On Random Graphs"<sup id="cite_ref-On_Random_Graphs_8-1" class="reference"><a href="#cite_note-On_Random_Graphs-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and independently by Gilbert in his paper "Random graphs".<sup id="cite_ref-Random_graphs_6-1" class="reference"><a href="#cite_note-Random_graphs-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Bose%E2%80%93Einstein_condensation%3A_a_network_theory_approach" class="mw-redirect" title="Bose–Einstein condensation: a network theory approach">Bose–Einstein condensation: a network theory approach</a></li>

<li><a href="Cavity_method" title="Cavity method">Cavity method</a> – Mathematical method in statistical physics</li>

<li><a href="Complex_networks" class="mw-redirect" title="Complex networks">Complex networks</a> – Network with non-trivial topological features<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Dual-phase_evolution" title="Dual-phase evolution">Dual-phase evolution</a> – Process that drives self-organization within complex adaptive systems</li>

<li><a href="Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a> – Two closely related models for generating random graphs</li>

<li><a href="Exponential_random_graph_model" class="mw-redirect" title="Exponential random graph model">Exponential random graph model</a></li>

<li><a href="Graph_theory" title="Graph theory">Graph theory</a> – Area of discrete mathematics</li>

<li><a href="Interdependent_networks" title="Interdependent networks">Interdependent networks</a> – Subfield of network science</li>

<li><a href="Network_science" title="Network science">Network science</a> – Academic field</li>

<li><a href="Percolation" title="Percolation">Percolation</a> – Filtration of fluids through porous materials</li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation theory</a> – Mathematical theory on behavior of connected clusters in a random graph</li>

<li><a href="Random_graph_theory_of_gelation" title="Random graph theory of gelation">Random graph theory of gelation</a> – Mathematical theory for sol–gel processes</li>

<li><a href="Regular_graph" title="Regular graph">Regular graph</a> – Graph where each vertex has the same number of neighbors</li>

<li><a href="Scale_free_network" class="mw-redirect" title="Scale free network">Scale free network</a> – Network whose degree distribution follows a power law<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Semilinear_response" title="Semilinear response">Semilinear response</a> – Extension of linear response theory in mesoscopic regimes</li>

<li><a href="Stochastic_block_model" title="Stochastic block model">Stochastic block model</a> – Concept in network science</li>

<li><a href="Lancichinetti%E2%80%93Fortunato%E2%80%93Radicchi_benchmark" title="Lancichinetti–Fortunato–Radicchi benchmark">Lancichinetti–Fortunato–Radicchi benchmark</a> – Algorithm<span style="display:none" class="category-spaceless-annotation">Pages displaying short descriptions with no spaces</span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBollobás2001" class="citation book cs1">Bollobás, Béla (2001). <i>Random Graphs</i> (2nd ed.). Cambridge University Press.</cite></span>

</li>

<li id="cite_note-Introduction_to_Random_graphs-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Introduction_to_Random_graphs_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFFriezeKaronski2015" class="citation book cs1">Frieze, Alan; Karonski, Michal (2015). <i>Introduction to Random Graphs</i>. Cambridge University Press.</cite></span>

</li>

<li id="cite_note-Random_Graphs2-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Random_Graphs2_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><a href="B%C3%A9la_Bollob%C3%A1s" title="Béla Bollobás">Béla Bollobás</a>, <i>Random Graphs</i>, 1985, Academic Press Inc., London Ltd.</span>

</li>

<li id="cite_note-Random_Graphs3-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Random_Graphs3_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Random_Graphs3_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Random_Graphs3_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="B%C3%A9la_Bollob%C3%A1s" title="Béla Bollobás">Béla Bollobás</a>, <i>Probabilistic Combinatorics and Its Applications</i>, 1991, Providence, RI: American Mathematical Society.</span>

</li>

<li id="cite_note-Handbook-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Handbook_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Handbook_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="B%C3%A9la_Bollob%C3%A1s" title="Béla Bollobás">Bollobas, B.</a> and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003</span>

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<li id="cite_note-Random_graphs-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Random_graphs_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Random_graphs_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGilbert1959" class="citation cs2"><a href="Edgar_Gilbert" title="Edgar Gilbert">Gilbert, E. N.</a> (1959), "Random graphs", <i>Annals of Mathematical Statistics</i>, <b>30</b> (4): <span class="nowrap">1141–</span>1144, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177706098">10.1214/aoms/1177706098</a></span></cite>.</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFNewman2010" class="citation book cs1">Newman, M. E. J. (2010). <i>Networks: An Introduction</i>. Oxford.</cite></span>

</li>

<li id="cite_note-On_Random_Graphs-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-On_Random_Graphs_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-On_Random_Graphs_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="Paul_Erd%C5%91s" title="Paul Erdős">Erdős, P.</a> <a href="Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Rényi, A</a> (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297 <a rel="nofollow" class="external autonumber" href="http://www.renyi.hu/~p_erdos/1959-11.pdf">[1]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200807021117/https://www.renyi.hu/~p_erdos/1959-11.pdf">Archived</a> 2020-08-07 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFRamezanpourKarimipourMashaghi2003" class="citation journal cs1">Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003). "Generating correlated networks from uncorrelated ones". <i>Phys. Rev. E</i>. <b>67</b> (46107): 046107. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0212469">cond-mat/0212469</a></span>. <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/2003PhRvE..67d6107R">2003PhRvE..67d6107R</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%2FPhysRevE.67.046107">10.1103/PhysRevE.67.046107</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/12786436">12786436</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:33054818">33054818</a>.</cite></span>

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<li id="cite_note-Chromatic_Polynomials_of_Random_Graphs-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chromatic_Polynomials_of_Random_Graphs_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFVan_BusselEhrlichFliegnerStolzenberg2010" class="citation journal cs1">Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc (2010). "Chromatic Polynomials of Random Graphs". <i>J. Phys. A: Math. Theor</i>. <b>43</b> (17): 175002. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1709.06209">1709.06209</a></span>. <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/2010JPhA...43q5002V">2010JPhA...43q5002V</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%2F43%2F17%2F175002">10.1088/1751-8113/43/17/175002</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:15723612">15723612</a>.</cite></span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFMorenoJennings1938" class="citation journal cs1">Moreno, Jacob L; Jennings, Helen Hall (Jan 1938). <a rel="nofollow" class="external text" href="https://hal.science/hal-03963403/file/morenojennings1938groupefmr.pdf">"Statistics of Social Configurations"</a> <span class="cs1-format">(PDF)</span>. <i>Sociometry</i>. <b>1</b> (3/4): <span class="nowrap">342–</span>374. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2785588">10.2307/2785588</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2785588">2785588</a>.</cite></span>

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<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFSolomonoffRapoport1951" class="citation journal cs1">Solomonoff, Ray; Rapoport, Anatol (June 1951). "Connectivity of random nets". <i>Bulletin of Mathematical Biophysics</i>. <b>13</b> (2): <span class="nowrap">107–</span>117. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02478357">10.1007/BF02478357</a>.</cite></span>

</li>

</ol></div></div>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Stochastic_processes496" style="padding:3px"><table class="nowraplinks 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="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Moran_process" title="Moran process">Moran process</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul><li><a href="Compound_Poisson_process" title="Compound Poisson process">Compound</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

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

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

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

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

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

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li><a href="Ising_model" title="Ising model">Ising model</a>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

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

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

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

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

</ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li>

<li><a href="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

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

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

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

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

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

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

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

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</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q910404#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1552" 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="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q910404#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1552" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Random graphs"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85111348">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Graphes aléatoires"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb12288320t">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Graphes aléatoires"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb12288320t">BnF data</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="náhodné grafy"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph664004&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007563131005171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/fbb1d70a-ca5f-4881-9c57-50968a42e355">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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