Hyperbolic geometry of complex network

Percolation in self-similar networksPRL 106:048701, 2011Dmitri KrioukovCAIDA/UCSDM. . Serrano, M. Bogu

NetSci, Budapest, June 20111PercolationPercolation is one of the most fundamental and best-studied critical phenomena in natureIn networks: the critical parameter is often average degree k, and there is kc such that:If k < kc: many small connected componentsIf k > kc: a giant connected component emergeskc can be zeroThe smaller the kc, the more robust is the network with respect to random damageAnalytic approachesUsually based on tree-like approximationsEmploying generating functions for branching processesOk, for zero-clustering networksConfiguration modelPreferential attachmentNot ok for strongly clustered networksReal networksNewman-Gleeson:Any target concentration of subgraphs, butThe network is tree-like at the subgraph levelReal networks:The distribution of the number of triangles overlapping over an edge is scale-freeIdentification of percolation universality classes of networksProblem is seemingly difficultDetails seem to prevailFew results are available for some networksConformal invarianceand percolationConformal invarianceand percolationJ. Cardys crossing formula:

, where

Proved by S. Smirnov (Fields Medal)

z1z2z3z4Scale invarianceand self-similaritySelf-similarityof network ensemblesLet ({}) be a network ensemble in the thermodynamic (continuum) limit, where {} is a set of its parameters (in ER, {} = k)Let be a rescaling transformationFor each graph G ({}), selects Gs subgraph G according to some ruleDefinition: ensemble ({}) is self-similar if ({}) = ({})where {} is some parameter transformation2210010001

Model propertiesVery general geometric network modelCan model networks with anyAverage degreePower-law degree distribution exponentClusteringSubsumes many popular random graph models as limiting regimes with degenerate geometriesHas a well-defined thermodynamic limitNodes cover all the hyperbolic planeWhat about rescaling?Rescaling transformation Very simple: just throw out nodes of degrees >

TheoremTheorem:Self-similar networks with growing k() and linear N (N) have zero percolation threshold (kc = 0)Proof:Suppose it is not true, i.e., kc > 0, and consider graph G below the threshold, i.e., a graph with k < kc which has no giant componentSince k() is growing, there exist such that Gs subgraph G is above the threshold, k > kc, i.e., G does have a giant componentContradiction: a graph that does not have a giant component contains a subgraph that doesConclusionsAmazingly simple proof of the strongest possible structural robustness for an amazingly general class of random networksApplies also to random graphs with given degree correlations, and to some growing networksSelf-similarity, or more generally, scale or conformal invariance, seem to be the pivotal properties for percolation in networksOther details appear unimportantConjecturing three obvious percolation universality classes for self-similar networksk() increases, decreases, or constant (PA!)The proof can be generalized to any processes whose critical parameters depend on k monotonicallyIsing, SIS models, etc.