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title="Graph drawing">Graph drawing</a></li> <li><a href="/wiki/Social_capital" title="Social capital">Social capital</a></li> <li><a href="/wiki/Link_analysis" title="Link analysis">Link analysis</a></li> <li><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Optimization</a></li> <li><a href="/wiki/Reciprocity_(network_science)" title="Reciprocity (network science)">Reciprocity</a></li> <li><a href="/wiki/Triadic_closure" title="Triadic closure">Closure</a></li> <li><a href="/wiki/Homophily" title="Homophily">Homophily</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitivity</a></li> <li><a href="/wiki/Preferential_attachment" title="Preferential attachment">Preferential attachment</a></li> <li><a href="/wiki/Balance_theory" title="Balance theory">Balance theory</a></li> <li><a href="/wiki/Network_effect" title="Network effect">Network effect</a></li> <li><a href="/wiki/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="/wiki/Computer_network" title="Computer network">Informational (computing)</a></li> <li><a href="/wiki/Telecommunications_network" title="Telecommunications network">Telecommunication</a></li> <li><a href="/wiki/Transport_network" class="mw-redirect" title="Transport network">Transport</a></li> <li><a href="/wiki/Social_network" title="Social network">Social</a></li> <li><a href="/wiki/Scientific_collaboration_network" title="Scientific collaboration network">Scientific collaboration</a></li> <li><a href="/wiki/Biological_network" title="Biological network">Biological</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural</a></li> <li><a href="/wiki/Interdependent_networks" title="Interdependent networks">Interdependent</a></li> <li><a href="/wiki/Semantic_network" title="Semantic network">Semantic</a></li> <li><a href="/wiki/Spatial_network" title="Spatial network">Spatial</a></li> <li><a href="/wiki/Dependency_network" title="Dependency network">Dependency</a></li> <li><a href="/wiki/Flow_network" title="Flow network">Flow</a></li> <li><a href="/wiki/Network_on_a_chip" title="Network on a chip">on-Chip</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/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;"> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Features</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Clique_(graph_theory)" title="Clique (graph theory)">Clique</a></li> <li><a href="/wiki/Connected_component_(graph_theory)" class="mw-redirect" title="Connected component (graph theory)">Component</a></li> <li><a href="/wiki/Cut_(graph_theory)" title="Cut (graph theory)">Cut</a></li> <li><a href="/wiki/Cycle_(graph_theory)" title="Cycle (graph theory)">Cycle</a></li> <li><a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Data structure</a></li> <li><a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">Edge</a></li> <li><a href="/wiki/Loop_(graph_theory)" title="Loop (graph theory)">Loop</a></li> <li><a href="/wiki/Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">Neighborhood</a></li> <li><a href="/wiki/Path_(graph_theory)" title="Path (graph theory)">Path</a></li> <li><a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">Vertex</a></li> <li><span class="nowrap"><a href="/wiki/Adjacency_list" title="Adjacency list">Adjacency list</a> / <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">matrix</a></span></li> <li><span class="nowrap"><a href="/wiki/Incidence_list" class="mw-redirect" title="Incidence list">Incidence list</a> / <a href="/wiki/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="/wiki/Bipartite_graph" title="Bipartite graph">Bipartite</a></li> <li><a href="/wiki/Complete_graph" title="Complete graph">Complete</a></li> <li><a href="/wiki/Directed_graph" title="Directed graph">Directed</a></li> <li><a href="/wiki/Hypergraph" title="Hypergraph">Hyper</a></li> <li><a href="/wiki/Labeled_graph" class="mw-redirect" title="Labeled graph">Labeled</a></li> <li><a href="/wiki/Multigraph" title="Multigraph">Multi</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random</a></li> <li><a href="/wiki/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="/wiki/Metrics_(networking)" title="Metrics (networking)">Metrics</a></li><li><a href="/wiki/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="/wiki/Centrality" title="Centrality">Centrality</a></li> <li><a href="/wiki/Degree_(graph_theory)" title="Degree (graph theory)">Degree</a></li> <li><a href="/wiki/Network_motif" title="Network motif">Motif</a></li> <li><a href="/wiki/Clustering_coefficient" title="Clustering coefficient">Clustering</a></li> <li><a href="/wiki/Degree_distribution" title="Degree distribution">Degree distribution</a></li> <li><a href="/wiki/Assortativity" title="Assortativity">Assortativity</a></li> <li><a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">Distance</a></li> <li><a href="/wiki/Modularity_(networks)" title="Modularity (networks)">Modularity</a></li> <li><a href="/wiki/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;"> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Topology</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi</a></li> <li><a href="/wiki/Barab%C3%A1si%E2%80%93Albert_model" title="Barabási–Albert model">Barabási–Albert</a></li> <li><a href="/wiki/Bianconi%E2%80%93Barab%C3%A1si_model" title="Bianconi–Barabási model">Bianconi–Barabási</a></li> <li><a href="/wiki/Fitness_model_(network_theory)" title="Fitness model (network theory)">Fitness model</a></li> <li><a href="/wiki/Watts%E2%80%93Strogatz_model" title="Watts–Strogatz model">Watts–Strogatz</a></li> <li><a href="/wiki/Exponential_random_graph_models" class="mw-redirect" title="Exponential random graph models">Exponential random (ERGM)</a></li> <li><a href="/wiki/Random_geometric_graph" title="Random geometric graph">Random geometric (RGG)</a></li> <li><a class="mw-selflink selflink">Hyperbolic (HGN)</a></li> <li><a href="/wiki/Hierarchical_network_model" title="Hierarchical network model">Hierarchical</a></li> <li><a href="/wiki/Stochastic_block_model" title="Stochastic block model">Stochastic block</a></li> <li><a href="/wiki/Blockmodeling" title="Blockmodeling">Blockmodeling</a></li> <li><a href="/wiki/Maximum-entropy_random_graph_model" title="Maximum-entropy random graph model">Maximum entropy</a></li> <li><a href="/wiki/Soft_configuration_model" title="Soft configuration model">Soft configuration</a></li> <li><a href="/wiki/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="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li> <li><a href="/wiki/Agent-based_model" title="Agent-based model">agent based</a></li> <li><a href="/wiki/Epidemic_model" class="mw-redirect" title="Epidemic model">Epidemic</a>/<a href="/wiki/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="/wiki/List_of_network_theory_topics" title="List of network theory topics">Topics</a></li> <li><a href="/wiki/Social_network_analysis_software" title="Social network analysis software">Software</a></li> <li><a href="/wiki/List_of_network_scientists" title="List of network scientists">Network scientists</a></li></ul> <ul><li><a href="/wiki/Category:Network_theory" title="Category:Network theory">Category:Network theory</a></li> <li><a href="/wiki/Category:Graph_theory" title="Category:Graph theory">Category:Graph theory</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Network_science" title="Template:Network science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Network_science" title="Template talk:Network science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Network_science" title="Special:EditPage/Template:Network science"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>hyperbolic geometric graph (HGG)</b> or <b>hyperbolic geometric network (HGN)</b> is a special type of <a href="/wiki/Spatial_network" title="Spatial network">spatial network</a> where (1) latent coordinates of nodes are sprinkled according to a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> into a <a href="/wiki/Hyperbolic_space" title="Hyperbolic space">hyperbolic space</a> of <a href="/wiki/Constant_function" title="Constant function">constant</a> negative <a href="/wiki/Constant_curvature" title="Constant curvature">curvature</a> and (2) an <a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">edge</a> between two nodes is present if they are close according to a function of the <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a><sup id="cite_ref-RefSpatialReview_1-0" class="reference"><a href="#cite_note-RefSpatialReview-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-RefDimaPre_2-0" class="reference"><a href="#cite_note-RefDimaPre-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (typically either a <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> resulting in deterministic connections between vertices closer than a certain threshold distance, or a decaying function of hyperbolic distance yielding the connection probability). A HGG generalizes a <a href="/wiki/Random_geometric_graph" title="Random geometric graph">random geometric graph</a> (RGG) whose embedding space is <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean.</a> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=1" title="Edit section: Mathematical formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematically, a <b>HGG</b> is a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> <span class="mwe-math-element"><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(V,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(V,E)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc43221ee885a10b87c89373b12b8c1110e7ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.233ex; height:2.843ex;" alt="{\displaystyle G(V,E)}" /></span> with a vertex <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>V</i> (<a href="/wiki/Cardinality" title="Cardinality">cardinality</a> <span class="mwe-math-element"><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=|V|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=|V|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7b2916b5edd1213ce832fb2aca91bff96c73ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.243ex; height:2.843ex;" alt="{\displaystyle N=|V|}" /></span>) and an edge set <i>E</i> constructed by considering the nodes as points placed onto a 2-dimensional hyperbolic space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{\zeta }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} _{\zeta }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cae0ff577910b1aefc196a447afb131fcb06ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.862ex; height:3.509ex;" alt="{\displaystyle \mathbb {H} _{\zeta }^{2}}" /></span> of constant negative <a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\zeta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\zeta ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a35764e66bdacf786b9b4c04fd7b88cee9e693a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.981ex; height:3.009ex;" alt="{\displaystyle -\zeta ^{2}}" /></span> and cut-off radius <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span>, i.e. the radius of the <a href="/wiki/Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk</a> which can be visualized using a <a href="/wiki/Hyperboloid_model" title="Hyperboloid model">hyperboloid model</a>. Each point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span> has hyperbolic polar coordinates <span class="mwe-math-element"><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_{i},\theta _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r_{i},\theta _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48a76a64e03fe87f46956ccde6239008999c7112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.582ex; height:2.843ex;" alt="{\displaystyle (r_{i},\theta _{i})}" /></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq r_{i}\leq R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r_{i}\leq R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8381629bb74cd906a4ff79c7119c132b8f310645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.972ex; height:2.509ex;" alt="{\displaystyle 0\leq r_{i}\leq R}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \theta _{i}<2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \theta _{i}<2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ffb3c84088fa450058328357369c0c35515136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.744ex; height:2.509ex;" alt="{\displaystyle 0\leq \theta _{i}<2\pi }" /></span>. </p><p>The <a href="/wiki/Hyperbolic_law_of_cosines" title="Hyperbolic law of cosines">hyperbolic law of cosines</a> allows to measure the distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0116346390c7dcd72d9ed714177a9b6c81e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.686ex; height:2.843ex;" alt="{\displaystyle d_{ij}}" /></span> between two points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" /></span>,<sup id="cite_ref-RefDimaPre_2-1" class="reference"><a href="#cite_note-RefDimaPre-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh(\zeta d_{ij})=\cosh(\zeta r_{i})\cosh(\zeta r_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ζ</mi> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ζ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ζ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cosh(\zeta d_{ij})=\cosh(\zeta r_{i})\cosh(\zeta r_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4c114dc8703fb0555017bcad5812419e148e7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.902ex; height:3.009ex;" alt="{\displaystyle \cosh(\zeta d_{ij})=\cosh(\zeta r_{i})\cosh(\zeta r_{j})}" /></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sinh(\zeta r_{i})\sinh(\zeta r_{j})\cos {\bigg (}\underbrace {\pi \!-\!{\bigg |}\pi -|\theta _{i}\!-\!\theta _{j}|{\bigg |}} _{\Delta }{\bigg )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ζ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ζ</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>π</mi> <mspace width="negativethinmathspace"></mspace> <mo>−</mo> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mi>π</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="negativethinmathspace"></mspace> <mo>−</mo> <mspace width="negativethinmathspace"></mspace> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sinh(\zeta r_{i})\sinh(\zeta r_{j})\cos {\bigg (}\underbrace {\pi \!-\!{\bigg |}\pi -|\theta _{i}\!-\!\theta _{j}|{\bigg |}} _{\Delta }{\bigg )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8104d553f50b1f817a43b3d68c75e57c9ad054fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:44.562ex; height:9.176ex;" alt="{\displaystyle -\sinh(\zeta r_{i})\sinh(\zeta r_{j})\cos {\bigg (}\underbrace {\pi \!-\!{\bigg |}\pi -|\theta _{i}\!-\!\theta _{j}|{\bigg |}} _{\Delta }{\bigg )}.}" /></span></dd></dl> <p>The angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" /></span> is the (smallest) angle between the two <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position vectors</a>. </p><p>In the simplest case, an edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}" /></span> is established iff (if and only if) two nodes are within a certain <i>neighborhood radius</i> <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" 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}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{ij}\leq r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>≤</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{ij}\leq r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8cf0ffa63e52e4ef164a730413e8cde71c16459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.833ex; height:2.843ex;" alt="{\displaystyle d_{ij}\leq r}" /></span>, this corresponds to an influence threshold. </p> <div class="mw-heading mw-heading3"><h3 id="Connectivity_decay_function">Connectivity decay function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=2" title="Edit section: Connectivity decay function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, a link will be established with a probability depending on the distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0116346390c7dcd72d9ed714177a9b6c81e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.686ex; height:2.843ex;" alt="{\displaystyle d_{ij}}" /></span>. A <i>connectivity decay function</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (s):\mathbb {R} ^{+}\to [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (s):\mathbb {R} ^{+}\to [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c517cd783fae5dafc68ffa4db664bef1f9488dfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.555ex; height:3.009ex;" alt="{\displaystyle \gamma (s):\mathbb {R} ^{+}\to [0,1]}" /></span> represents the probability of assigning an edge to a pair of nodes at distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" /></span>. In this framework, the simple case of <i><a href="/wiki/Depletion_force#Hard-sphere_potential" title="Depletion force">hard-code</a></i> neighborhood like in <a href="/wiki/Random_geometric_graph" title="Random geometric graph">random geometric graphs</a> is referred to as <i>truncation decay function</i>.<sup id="cite_ref-RefConnectivityDecay_3-0" class="reference"><a href="#cite_note-RefConnectivityDecay-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generating_hyperbolic_geometric_graphs">Generating hyperbolic geometric graphs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=3" title="Edit section: Generating hyperbolic geometric graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Krioukov et al.<sup id="cite_ref-RefDimaPre_2-2" class="reference"><a href="#cite_note-RefDimaPre-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> describe how to generate hyperbolic geometric graphs with uniformly random node distribution (as well as generalized versions) on a disk of radius <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{\zeta }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} _{\zeta }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cae0ff577910b1aefc196a447afb131fcb06ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.862ex; height:3.509ex;" alt="{\displaystyle \mathbb {H} _{\zeta }^{2}}" /></span>. These graphs yield a <a href="/wiki/Power-law_distribution" class="mw-redirect" title="Power-law distribution">power-law distribution</a> for the node degrees. The angular coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> of each point/node is chosen uniformly random from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,2\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348d40bf3f8b7e1c00c4346440d7e2e4f0cc9b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.985ex; height:2.843ex;" alt="{\displaystyle [0,2\pi ]}" /></span>, while the density function for the radial coordinate r is chosen according to the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" /></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (r)=\alpha {\frac {\sinh(\alpha r)}{\cosh(\alpha R)-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>R</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (r)=\alpha {\frac {\sinh(\alpha r)}{\cosh(\alpha R)-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c82e15e92fe827f1e4386e0521cf612493866e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.95ex; height:6.509ex;" alt="{\displaystyle \rho (r)=\alpha {\frac {\sinh(\alpha r)}{\cosh(\alpha R)-1}}}" /></span></dd></dl> <p>The growth parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}" /></span> controls the distribution: For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc830c1d815ac7d39093030e335fc3154b37cc00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.681ex; height:2.509ex;" alt="{\displaystyle \alpha =\zeta }" /></span>, the distribution is uniform in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{\zeta }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} _{\zeta }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cae0ff577910b1aefc196a447afb131fcb06ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.862ex; height:3.509ex;" alt="{\displaystyle \mathbb {H} _{\zeta }^{2}}" /></span>, for smaller values the nodes are distributed more towards the center of the disk and for bigger values more towards the border. In this model, edges between nodes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" /></span> exist iff <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{uv}<R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo><</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{uv}<R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32fca779857b8894c64b38ead629fcee6f9c8e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.041ex; height:2.509ex;" alt="{\displaystyle d_{uv}<R}" /></span> or with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (d_{uv})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (d_{uv})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f64c53dcdf801c1bd1fd1a72aafddc0a8ed0d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle \gamma (d_{uv})}" /></span> if a more general connectivity decay function is used. The average degree is controlled by the radius <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> of the hyperbolic disk. It can be shown, that for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha /\zeta >1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ζ</mi> <mo>></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha /\zeta >1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380935be5f016a8725a16ee244fda69f70cf834c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.331ex; height:2.843ex;" alt="{\displaystyle \alpha /\zeta >1/2}" /></span> the node degrees follow a power law distribution with exponent <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1+2\alpha /\zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =1+2\alpha /\zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5df04c916a920f5f10ead9f4a1ce692d7da1b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.271ex; height:2.843ex;" alt="{\displaystyle \gamma =1+2\alpha /\zeta }" /></span>. </p><p> The image depicts randomly generated graphs for different values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> and <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} _{1}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1d81ce5943f66ba48f2f98752640dc9e889a6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.862ex; height:3.176ex;" alt="{\displaystyle \mathbb {H} _{1}^{2}}" /></span>. It can be seen how <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> has an effect on the distribution of the nodes and <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> on the connectivity of the graph. The native representation where the distance variables have their true hyperbolic values is used for the visualization of the graph, therefore edges are straight lines. </p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Hyperbolic_graphs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_graphs.png/600px-Hyperbolic_graphs.png" decoding="async" width="600" height="393" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_graphs.png/900px-Hyperbolic_graphs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/89/Hyperbolic_graphs.png 2x" data-file-width="970" data-file-height="636" /></a><figcaption>Random hyperbolic geometric graphs with N=100 nodes each for different values of alpha and R</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Quadratic_complexity_generator">Quadratic complexity generator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=4" title="Edit section: Quadratic complexity generator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Source:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The naive algorithm for the generation of hyperbolic geometric graphs distributes the nodes on the hyperbolic disk by choosing the angular and radial coordinates of each point are sampled randomly. For every pair of nodes an edge is then inserted with the probability of the value of the connectivity decay function of their respective distance. The pseudocode looks as follows: </p> <pre><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\{\},E=\{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\{\},E=\{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878073633f339850adf83f1256368e52ac8c3840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.444ex; height:2.843ex;" alt="{\displaystyle V=\{\},E=\{\}}" /></span> <b>for</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\gets 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">←</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\gets 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83306299648e80b8f73280df6fb3e7d2e000b1ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.579ex; height:2.176ex;" alt="{\displaystyle i\gets 0}" /></span> <b>to</b> <span class="mwe-math-element"><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-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86aeb216b214f70df1341f34ce273cd3582ce2aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.066ex; height:2.343ex;" alt="{\displaystyle N-1}" /></span> <b>do</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \gets U[0,2\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo stretchy="false">←</mo> <mi>U</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta \gets U[0,2\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc8612fded049bc692f75795727f5e5aad0daeab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.472ex; height:2.843ex;" alt="{\displaystyle \theta \gets U[0,2\pi ]}" /></span> <span class="mwe-math-element"><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\gets {\frac {1}{\alpha }}{\text{acosh}}(1+(\cosh \alpha R-1)U[0,1])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">←</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>acosh</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>α</mi> <mi>R</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>U</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\gets {\frac {1}{\alpha }}{\text{acosh}}(1+(\cosh \alpha R-1)U[0,1])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bceaae145512c1a462ca19969d44b7d38a59a10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.654ex; height:5.176ex;" alt="{\displaystyle r\gets {\frac {1}{\alpha }}{\text{acosh}}(1+(\cosh \alpha R-1)U[0,1])}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V\cup \{(r,\theta )\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>V</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=V\cup \{(r,\theta )\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd60f117076a3f532f880107dc07285d9244f935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.563ex; height:2.843ex;" alt="{\displaystyle V=V\cup \{(r,\theta )\}}" /></span> <b>for every pair </b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)\in V\times V,u\neq v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>V</mi> <mo>×</mo> <mi>V</mi> <mo>,</mo> <mi>u</mi> <mo>≠</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u,v)\in V\times V,u\neq v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2342b94d2eb18464f7be3999fb2ea886418953d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.146ex; height:2.843ex;" alt="{\displaystyle (u,v)\in V\times V,u\neq v}" /></span> <b>do</b> <b>if</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U[0,1]\leq \gamma (d_{uv})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>≤</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U[0,1]\leq \gamma (d_{uv})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06d82e17a0aeeefd0764e7c79b471f6866d0cad9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.784ex; height:2.843ex;" alt="{\displaystyle U[0,1]\leq \gamma (d_{uv})}" /></span> <b>then</b> <span class="mwe-math-element"><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=E\cup \{(u,v)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>E</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E\cup \{(u,v)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb71dbfc1ba34514cae277a0bfa3bc95fefa56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.858ex; height:2.843ex;" alt="{\displaystyle E=E\cup \{(u,v)\}}" /></span> <b>return</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V,E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>,</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e779fc22e4863935337975b745a1bea4b0547b03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.597ex; height:2.509ex;" alt="{\displaystyle V,E}" /></span> </pre> <p><span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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}" /></span> is the number of nodes to generate, the distribution of the radial coordinate by the probability density function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" /></span> is achieved by using <a href="/wiki/Inverse_transform_sampling" title="Inverse transform sampling">inverse transform sampling</a>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></span>denotes the uniform sampling of a value in the given interval. Because the algorithm checks for edges for all pairs of nodes, the runtime is quadratic. For applications where <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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}" /></span> is big, this is not viable any more and algorithms with subquadratic runtime are needed. </p> <div class="mw-heading mw-heading3"><h3 id="Sub-quadratic_generation">Sub-quadratic generation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=5" title="Edit section: Sub-quadratic generation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To avoid checking for edges between every pair of nodes, modern generators use additional <a href="/wiki/Data_structure" title="Data structure">data structures</a> that <a href="/wiki/Graph_partition" title="Graph partition">partition</a> the graph into bands.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A visualization of this shows a hyperbolic graph with the boundary of the bands drawn in orange. In this case, the partitioning is done along the radial axis. Points are stored sorted by their angular coordinate in their respective band. For each point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /></span>, the limits of its hyperbolic circle of radius <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></span> can be (over-)estimated and used to only perform the edge-check for points that lie in a band that intersects the circle. Additionally, the sorting within each band can be used to further reduce the number of points to look at by only considering points whose angular coordinate lie in a certain range around the one of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /></span> (this range is also computed by over-estimating the hyperbolic circle around <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" /></span>). </p><p>Using this and other extensions of the algorithm, time complexities of <span class="mwe-math-element"><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 {O}}(n\log \log n+m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n\log \log n+m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65145c690086650402a8a169d45bf7e731d8c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.434ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(n\log \log n+m)}" /></span>(where <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}" /></span> is the number of nodes and <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" 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}" /></span> the number of edges) are possible with high probability.<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> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Partitioning_of_a_hyperbolic_geometric_graph_into_bands.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Partitioning_of_a_hyperbolic_geometric_graph_into_bands.png/349px-Partitioning_of_a_hyperbolic_geometric_graph_into_bands.png" decoding="async" width="349" height="349" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/67/Partitioning_of_a_hyperbolic_geometric_graph_into_bands.png 1.5x" data-file-width="404" data-file-height="404" /></a><figcaption>The hyperbolic graph is partitioned into bands such that each of them holds approximately the same number of points.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Findings">Findings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=6" title="Edit section: Findings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63f7f0a961df774bd3db35e02af655591ad23c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.356ex; height:2.509ex;" alt="{\displaystyle \zeta =1}" /></span> (Gaussian curvature <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aae3f03866ce5510e82dcd92d71af36a81d9204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.135ex; height:2.343ex;" alt="{\displaystyle K=-1}" /></span>), HGGs form an <a href="/wiki/Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">ensemble</a> of networks for which is possible to express the <a href="/wiki/Degree_distribution" title="Degree distribution">degree distribution</a> <a href="/wiki/Closed-form_expression" title="Closed-form expression">analytically as closed form</a> for the <a href="/wiki/Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting case</a> of large number of nodes.<sup id="cite_ref-RefDimaPre_2-3" class="reference"><a href="#cite_note-RefDimaPre-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This is worth mentioning since this is not true for many ensembles of graphs. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=7" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>HGGs have been suggested as promising model for <a href="/wiki/Social_networks" class="mw-redirect" title="Social networks">social networks</a> where the hyperbolicity appears through a competition between <i>similarity</i> and <i>popularity</i> of an individual.<sup id="cite_ref-RefPapaNature_8-0" class="reference"><a href="#cite_note-RefPapaNature-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbolic_geometric_graph&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li 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"Popularity versus similarity in growing networks". <i>Nature</i>. <b>489</b> (7417): <span class="nowrap">537–</span>540. <a href="/wiki/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/1106.0286">1106.0286</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012Natur.489..537P">2012Natur.489..537P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature11459">10.1038/nature11459</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22972194">22972194</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4424179">4424179</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Popularity+versus+similarity+in+growing+networks&rft.volume=489&rft.issue=7417&rft.pages=%3Cspan+class%3D%22nowrap%22%3E537-%3C%2Fspan%3E540&rft.date=2012-09-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4424179%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2012Natur.489..537P&rft_id=info%3Aarxiv%2F1106.0286&rft_id=info%3Apmid%2F22972194&rft_id=info%3Adoi%2F10.1038%2Fnature11459&rft.aulast=Papadopoulos&rft.aufirst=Fragkiskos&rft.au=Kitsak%2C+Maksim&rft.au=Serrano%2C+M.+%C3%81ngeles&rft.au=Bogu%C3%B1%C3%A1%2C+Mari%C3%A1n&rft.au=Krioukov%2C+Dmitri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHyperbolic+geometric+graph" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hyperbolic_geometric_graph&oldid=1265659971">https://en.wikipedia.org/w/index.php?title=Hyperbolic_geometric_graph&oldid=1265659971</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Networks" title="Category:Networks">Networks</a></li><li><a href="/wiki/Category:Network_theory" title="Category:Network theory">Network theory</a></li><li><a href="/wiki/Category:Geometric_graphs" title="Category:Geometric graphs">Geometric graphs</a></li><li><a href="/wiki/Category:Hyperbolic_geometry" title="Category:Hyperbolic geometry">Hyperbolic geometry</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden category: <ul><li><a href="/wiki/Category:CS1_errors:_missing_periodical" title="Category:CS1 errors: missing periodical">CS1 errors: missing periodical</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 28 December 2024, at 01:11<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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