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Hyperbolic Geometry of Complex Network

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Hyperbolic Geometry of Complex Network Duality between static and dynamic networks Dmitri Krioukov Massimo Ostilli CAIDA/UCSD NIST-Bell Labs Workshop on Large-Scale Networks October 2013 Motivation • Real networks are growing, but static network models exist and are widely used – Simpler – More tractable • In a recent work, a minimalistic growing extension of a “very static” model describes remarkably well the growth dynamics of some real networks (F. Papadopoulos et al., Nature, 489:537, 2012) • Informally: – One camp: there cannot be any connection – Another camp: there should be some connection • Preferential attachment versus configuration model: what is the difference??? Graph ensemble • Set of graphs 풢 • With probability measure 푃 퐺 , 푃(퐺) = 1 퐺∈풢 Random graphs with hidden variables • Given N nodes • Assign to each node a 휅푠 휅푡 random variable 휅 푝푁(휅푠, 휅푡) drawn from distribution 휌푁 휅 • Connect each node pair 푠, 푡 with probability 푝푁 휅푠, 휅푡 Example: Classical random graphs • Given N nodes • Hidden variables: None 휅 휅 • Connection 푠 푝 푡 probability: 푝푁 휅푠, 휅푡 = 푝 • Results: – Degree distribution: Poisson (휆 = 푝푁) – Clustering: Weak Example: Soft configuration model • Given N nodes • Hidden variables: Expected degrees 휅 휅푠휅푡 −훾 휅 푝 ~ 휅 휌 휅 ~ 휅 푠 푠푡 푁 푡 • Connection probability: 휅 휅 푝 휅 , 휅 = 푠 푡 푁 푠 푡 푘 푁 • Results: – Degree distribution: Power law (푃 푘 ~ 푘−훾) – Clustering: Weak Example: Random geometric graphs • Given N nodes • Hidden variables: Node coordinates 휅 in a space 휌 휅 : uniform in the space 푁 휅푠 휅푡 • Connection probability: 푑 휅푠, 휅푡 < 푅 푝푁 휅푠, 휅푡 = Θ 푅 − 푑푠푡 • Results: – Clustering: Strong – Degree distribution: • Euclidean or spherical space: Poisson • Hyperbolic or de Sitter spaces: Power law Probability measure • Let – 퐺 = 푎푠푡 the adjacency matrix – 휅 = 휅1, 휅2, … , 휅푁 a hidden variable assignment – 푝푠푡 = 푝 휅푠, 휅푡 the connection probability matrix • Then 푎푠푡 1−푎푠푡 – 푃 퐺 휅 = 푠<푡 푝푠푡 1 − 푝푠푡 푁 – 휌푁 휅 = 푡=1 휌푁 휅푡 – 푃 퐺 = ∫ 푃 퐺 휅 휌푁 휅 푑휅 Static versus dynamic graphs with hidden variables • Static ensemble 풢푆 • Dynamic ensemble 풢퐷 – Given N nodes – For each new node 푡 = 1,2, … – Assign to each node – Assign to 푡 a random 푡 = 1,2, … , 푁 a random variable 휅푡 drawn from variable 휅푡 drawn from distribution 휌푡 휅푡 distribution 휌푁 휅푡 – Connect each node pair – Connect 푡 to each existing 푠, 푡 with probability node 푠 with probability 푝푁 휅푠, 휅푡 푝푡 휅푠, 휅푡 Equivalence • Let dynamic 푡 = 1,2, … , 푁 • Two ensembles 풢푆 and 풢퐷 are identical (풢푆 = 풢퐷) if 푃푆 퐺 = 푃퐷 퐺 for all 퐺 ∈ 풢 • Distributions 푃푆 퐺 = 푃퐷 퐺 if – 휌푆 휅 = 휌퐷 휅 and – 푝푆 휅푠, 휅푡 = 푝퐷 휅푠, 휅푡 • The difference is – 휌푁 휅푡 (static) versus 휌푡 휅푡 (dynamic) – 푝푁 휅푠, 휅푡 (static) versus 푝푡 휅푠, 휅푡 (dynamic) Weak equivalence • Ensembles 풢푆 and 풢퐷 are weakly equivalent if 풢푆 = 풢퐷 for some 푁 • The simplest example – Dynamic 휌푡 휅푡 is equal to static 휌푁 휅푡 – Dynamic 푝푡 휅푠, 휅푡 is equal to static 푝푁 휅푠, 휅푡 – The difference is only in node labeling Strong equivalence • Ensembles 풢푆 and 풢퐷 are strongly equivalent if 풢푆 = 풢퐷 for any 푁 • In this case the connection probability cannot depend on graph size 푁 – 푝푁 휅푠, 휅푡 = 푝푡 휅푠, 휅푡 = 푝 휅푠, 휅푡 • The simplest example – 휌푁 휅푡 = 휌푡 휅푡 = 휌 휅푡 – Graphs are dense because average degree 푘 = 푁 ∬ 휌 휅 푝 휅, 휅′ 휌 휅′ 푑휅 푑휅′ = 푂 푁 • In sparse strongly equivalent ensembles, the connection probability does not depend on graph size, but the hidden variable distribution does Example: Classical random graphs • Ensemble 풢푁,푝 is strongly equivalent and dense since 푘 = 푁푝 – The growing definition is: connect new nodes 푡 to existing nodes 푠 < 푡 with probability 푝 • An attempt to fix: ensemble 풢푁,푘 : 푘 – Static: connect all node pairs with probability 푝 = 푁 • The ensemble is 풢푁,푝 • The degree distribution is Poisson – Dynamic: connect new nodes 푡 to existing nodes 푠 < 푡 with 푘 probability 푝 = 푡 • The ensemble is not 풢푁,푝 • The degree distribution is exponential • The ensembles are very different because the connection probability depends on graph size Example: Soft configuration model • Strong equivalence in the dense case (graphons) • No strong equivalence in the sparse case since 휅 휅 the connection probability 푝 ~ 푠 푡 depends 푠푡 푁 on graph size Example: Preferential attachment • Weak equivalence in the soft formulation (M.Boguna et al., PRE 68:36112, 2003) • No strong equivalence since the probability that new node 푡 connects to existing node 푠 푘푠 푝푠푡~ depends on graph size 푠 푘푠 • Needless to say, 푃 퐺 is unknown Example: Random geometric graphs • Strong equivalence in sparse graphs! • Connection probability 푝푠푡 = Θ 푅 − 푑푠푡 does not depend on graph size • The distribution of new node coordinates 휌푡 휅푡 is such that 휌푆 휅 = 휌퐷 휅 because both static and dynamic constructions implement the same (Poisson) point process Poisson point processes behind random geometric graphs • Let 푟푡 be the radial coordinate of node 푡 • Then 푉푡 = 퐵 0, 푟푡 is its volume coordinate • Points 푉푡, 푡 = 1,2, … is a PPP on ℝ+ Two pairs of identical ensembles • Fixed-volume pair – Static ensemble 풢푆,푉 • Given 푉, sample 푁 from the Poisson distribution with mean 훿푉, where 훿 is the point density (PPP rate) • Sample 푁 points on [0, 푉] – Dynamic ensemble 풢퐷,푉 • Given Δ푉, sample 푁 from the Poisson distribution with mean 훿Δ푉, where 훿 is the point density (PPP rate) • Sample 푁 points on [푉, 푉 + Δ푉] • Fixed-size pair – Static ensemble 풢푆,푁 • Given 푁, sample 푉 from the Gamma distribution with rate 훿 and shape 푁 • Sample 푁 − 1 points on [0, 푉] – Dynamic ensemble 풢퐷,푁 • Given ΔN, sample Δ푉 from the Gamma distribution with rate 훿 and shape Δ푁 • Sample Δ푁 − 1 points on [푉, 푉 + Δ푉] Two pairs of identical ensembles Final example: de Sitter causal sets • Random geometric graphs in de Sitter spacetime – Angular coordinates 휃 are as in all spherically symmetric random geometric graphs, i.e., sampled from the uniform distribution on a unit sphere 핊푑 – “Radial” (conformal time) coordinates 휂 are sampled from 휌 휂 ∼ sec푑+1 휂 – Nodes are connected if Δ휂 > Δ휃 • The resulting graph ensembles are – Strongly equivalent – Sparse – Have power-law distributions of node degrees with exponent 훾 = 2 – Have strong clustering Conclusions • There does exist a very simple connection between equilibrium and non-equilibrium approaches to network analysis • Some real networks appear to be “close” to graph ensembles that admit dual, equilibrium and growing, description • This justifies applications of powerful equilibrium tools to the analysis of real networks • Does nature optimize? • If the least action principle applies, then what is the action? • D. Krioukov and M. Ostilli Duality between equilibrium and growing networks, PRE, v.88, p.022808, 2013 .
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