SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Sparse network</title>

</head>

Sparse network

</h1>

</header>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Network_science" title="Network science">network science</a>, a sparse network has much fewer links than the possible maximum number of links within that network (the opposite is a dense network). The study of sparse networks is a relatively new area primarily stimulated by the study of real networks, such as social and computer networks.<a href="#cite_note-:0-1">[1]</a>

The notion of much fewer links is, of course, colloquial and informal. While a threshold for a particular network may be invented, there is no universal threshold that defines what much fewer actually means. As a result, there is no formal sense of sparsity for any finite network, despite widespread agreement that most empirical networks are indeed sparse. There is, however, a formal sense of sparsity in the case of infinite network models, determined by the behavior of the number of edges (M) and/or the average degree (⟨k⟩) as the number of nodes (N) goes to infinity.<a href="#cite_note-newman-2">[2]</a>

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

A simple unweighted network of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> in it is much smaller than the maximum possible number of links <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{max}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c60c6220c155da2b6486fd364309ae961fe588d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.739ex; height:2.509ex;" alt="{\displaystyle M_{max}}" loading="lazy">:<a href="#cite_note-:0-1">[1]</a>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\ll M_{max}={N \choose 2}}</annotation>

</semantics>

</math><img src="./ee31f0eb25b1ece080fa522ea70f13383c8c3936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.378ex; height:6.176ex;" alt="{\displaystyle M\ll M_{max}={N \choose 2}}" loading="lazy"> .

In any given (real) network, the number of nodes N and links M are just two numbers, therefore the meaning of the much smaller sign (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ll }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./10563f31a4178c71bef006ea5c601e908f984136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \ll }" loading="lazy"> above) is purely colloquial and informal, and so are statements like "many real networks are sparse."

However, if we deal with a synthetic graph sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy">, or a network model that is well defined for networks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"> of any size N = 1,2,...,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">, then the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ll }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./10563f31a4178c71bef006ea5c601e908f984136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \ll }" loading="lazy"> attains its usual formal meaning:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\ll M_{max}\iff M=o(M_{max})\iff \lim _{N\rightarrow \infty }{\frac {M}{M_{max}}}=0}</annotation>

</semantics>

</math><img src="./4cb2bb65e8c2617252973b2636dbf66596a13190.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.775ex; height:5.676ex;" alt="{\displaystyle M\ll M_{max}\iff M=o(M_{max})\iff \lim _{N\rightarrow \infty }{\frac {M}{M_{max}}}=0}" loading="lazy">.

In other words, a network sequence or model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"> is called dense or sparse depending on whether the (expected) average degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle =2M/N}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65f4de99fff8314be2a279d223e551ba3b16db0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.95ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle =2M/N}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"> scales linearly or sublinearly with N:<a href="#cite_note-newman-2">[2]</a><a href="#cite_note-bela-3">[3]</a>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"> is dense if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle =O(N)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e7e334fd73307f4e1def01b6bc7525521cb328a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.765ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle =O(N)}" loading="lazy">;

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"> is sparse if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle =o(N)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./de678b0b0810919cfb9b9684e040b71764d95a8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.12ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle =o(N)}" loading="lazy">.

An important subclass of sparse networks are networks whose average degree is either constant or converges to a constant. Some authors call only such networks sparse, while others reserve special names for them: <a href="#cite_note-janson-4">[4]</a>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"> is truly sparse or extremely sparse or ultrasparse if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle =O(1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle k\rangle =O(1)}</annotation>

</semantics>

</math><img src="./f2c9921d5f99ada84b60e4188e459c61550982bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.864ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle =O(1)}" loading="lazy">.

There also exist alternative, stricter definitions of network sparsity requiring the convergence of the degree distribution in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e784915eb57f908dc0064588f9d41ab93ff5bbaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.518ex; height:2.509ex;" alt="{\displaystyle G_{N}}" loading="lazy"> to a well defined limit at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\rightarrow \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3c6337428441dbe11ca917f0d9115e8cf3a78f40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.001ex; height:2.176ex;" alt="{\displaystyle N\rightarrow \infty }" loading="lazy">.<a href="#cite_note-remco-5">[5]</a> According to this definition, the <a href="Star_(graph_theory)" title="Star (graph theory)">N-star graph</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aca805f5a6c6548a825d119230ab751152321ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.116ex; height:2.509ex;" alt="{\displaystyle S_{N}}" loading="lazy">, for example, is not sparse.

<div class="mw-heading mw-heading2"><h2 id="Node_degree_distribution">Node degree distribution</h2></div>

The node degree distribution changes with the increasing connectivity. Different link densities in the complex networks have different node-degree distribution, as Flickr Network Analysis suggests.<a href="#cite_note-Scholz2015-6">[6]</a> The sparsely connected networks have a scale free, <a href="Power_law" title="Power law">power law</a> distribution. With increasing connectivity, the networks show increasing divergence from power law. One of the main factors, influencing on the network connectivity is the <a href="Node_(networking)" title="Node (networking)">node similarity</a>. For instance, in <a href="Social_networks" class="mw-redirect" title="Social networks">social networks</a>, people are likely to be linked to each other if they share common social background, interests, tastes, beliefs, etc. In context of biological networks, proteins or other molecules are linked if they have exact or complementary fit of their complex surfaces.<a href="#cite_note-Scholz2015-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Common_terminology">Common terminology</h2></div>

If the nodes in the networks are not weighted, the structural components of the network can be shown through <a href="Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a>. If the most elements in the matrix are zero, such matrix is referred as <a href="Sparse_matrix" title="Sparse matrix">sparse matrix</a>. In contrast, if most of the elements are nonzero, then the matrix is <a href="Dense_matrix" class="mw-redirect" title="Dense matrix">dense</a>. The sparsity or density of the matrix is identified by the fraction of the zero element to the total number of the elements in the matrix. Similarly, in the context of <a href="Graph_theory" title="Graph theory">graph theory</a>, if the number of links is close to its maximum, then the graph would be known as <a href="Dense_graph" title="Dense graph">dense graph</a>. If the number of links is lower than the maximum number of links, this type of graphs are referred as <a href="Sparse_graph" class="mw-redirect" title="Sparse graph">sparse graph</a>.<a href="#cite_note-7">[7]</a>

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

Sparse Network can be found in <a href="Social_network" title="Social network">social</a>, <a href="Computer_network" title="Computer network">computer</a> and <a href="Biological_network" title="Biological network">biological networks</a>, as well as, its applications can be found in <a href="Transport_network" class="mw-redirect" title="Transport network">transportation</a>, power-line, citation networks, etc. Since most real networks are large and sparse, there were several models developed to understand and analyze them.<a href="#cite_note-8">[8]</a> These networks have inspired sparse <a href="Network_on_a_chip" title="Network on a chip">network-on-chip</a> design in multiprocessor embedded <a href="Computer_engineering" title="Computer engineering">computer engineering</a>.

Sparse networks also induce cheaper computations by making it efficient to store the network as an <a href="Adjacency_list" title="Adjacency list">Adjacency list</a>, rather than an <a href="Adjacency_matrix" title="Adjacency matrix">Adjacency matrix</a>. For example, when using an adjacency list, iterating over a node's neighbors can be achieved in O(M/N), whereas it is achieved in O(N) with an adjacency matrix.<a href="#cite_note-newman-2">[2]</a>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-columns references-column-width reflist-columns-2">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBarabási2015" class="citation book cs1">Barabási, Albert-László (2015). <a rel="nofollow" class="external text" href="http://barabasi.com/networksciencebook/">Network Science</a>. Cambridge University Press. Retrieved 25 May 2015.</cite>

</li>

<li id="cite_note-newman-2">^ <a href="#cite_ref-newman_2-0">a</a> <a href="#cite_ref-newman_2-1">b</a> <a href="#cite_ref-newman_2-2">c</a> <cite id="CITEREFNewman" class="citation book cs1">Newman, Mark. <a rel="nofollow" class="external text" href="https://global.oup.com/academic/product/networks-9780198805090?cc=us&lang=en&">Networks 2nd Edition</a>. Retrieved 14 Feb 2021.</cite>

</li>

<li id="cite_note-bela-3"><a href="#cite_ref-bela_3-0">^</a> <cite id="CITEREFBollobás1985" class="citation book cs1">Bollobás, Béla (1985). Random Graphs. Academic Press.</cite>

</li>

<li id="cite_note-janson-4"><a href="#cite_ref-janson_4-0">^</a> <cite id="CITEREFJanson2018" class="citation journal cs1">Janson, Svante (2018). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6405020">"On Edge Exchangeable Random Graphs"</a>. J Stat Phys. 173 (3–4): 448–484. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1702.06396">1702.06396</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018JSP...173..448J">2018JSP...173..448J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10955-017-1832-9">10.1007/s10955-017-1832-9</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6405020">6405020</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/30930480">30930480</a>.</cite>

</li>

<li id="cite_note-remco-5"><a href="#cite_ref-remco_5-0">^</a> <cite id="CITEREFvan_der_Hofstad2017" class="citation book cs1">van der Hofstad, Remco (2017). Random Graphs and Complex Networks. Cambridge University Press. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781316779422">10.1017/9781316779422</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781316779422</bdi>.</cite>

</li>

<li id="cite_note-Scholz2015-6">^ <a href="#cite_ref-Scholz2015_6-0">a</a> <a href="#cite_ref-Scholz2015_6-1">b</a> <cite id="CITEREFScholz2015" class="citation journal cs1">Scholz, Matthias (7 January 2015). <a rel="nofollow" class="external text" href="http://jdmdh.episciences.org/77/pdf">"Node similarity as a basic principle behind connectivity in complex networks"</a>. Journal of Data Mining and Digital Humanities. 2015 (77). <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1010.0803">1010.0803</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.46298%2Fjdmdh.33">10.46298/jdmdh.33</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:221799">221799</a>. Retrieved 25 May 2015.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFNykamp" class="citation web cs1">Nykamp, Duane Q. <a rel="nofollow" class="external text" href="http://mathinsight.org/network_introduction">"An introduction to networks"</a>. Math Insight. Retrieved 25 May 2015.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFGribonval" class="citation web cs1">Gribonval, Rémi. <a rel="nofollow" class="external text" href="http://www.small-project.eu/">"Sparse Models, Algorithms and Learning for Large-scale data"</a>. SMALL. Retrieved 25 May 2015.</cite>

</li>

</ol></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-01-04" href="https://en.wikipedia.org/wiki/?title=Sparse_network&oldid=1193598850">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>