The Universal Lawsof Structural Dynamicsin Large GraphsDmitri KrioukovUCSD/CAIDA

David Meyer & David RideoutUCSD/Math

F. Papadopoulos, M. Kitsak, M. . Serrano, M. Bogu

M. Ostilli

DARPAs GRAPHS, Washington, DC, Halloween 2012(Cancelled by the Frankenstorm)c1High-level project descriptionMotivation:Predict network dynamicsDetect anomaliesGoal:Identify the universal laws of network dynamicsMethods: geometry: random geometric graphsPast work: static graphsPresent/future work: dynamic graphsOutlineHyperbolic popular similarGrowing random hyperbolic graphsNext stepRandom Lorentzian graphsRandom geometric graphDiscretization of a smooth manifolds (B.Riemann, Nature, v.7)

Take a circle of radius R

Sprinkle N points into it uniformly at random

Connect each pair of points iff the distance between them is x r RhyperbolichyperbolicR grows to R+dR1 new point in [R,R+dR]new point to existingGrowing

Connecting to m closest nodesThe expected distance to the mth closest node from t is:New node t located at radial coordinate rt ln t,and connecting to all nodes within distance Rt ~ rt,connects to a fixed number of closest nodesClosest nodesNew node t connects to a fixed number of existing nodes s with smallest sstThe hyperbolic distance between s and t isFind m nodes s, st, with smallest xst for a given t:Hyperbolic popular similarTwo dimensions of attractivenessRadial popularity: birth time s:The smaller the s, the more popular is the node sAngular similarity: distance st:The smaller the st, the more similar is the node s to tNew node t connects to existing nodes s optimizing trade-offs betweenpopularity and similarityThis trade-off optimization yieldshyperbolic geometryWhat else it yieldsPower-law graphsWith strongest possible clusteringEffective preferential attachment

ClusteringProbability of new connections from t to s so far

If we smoothen the threshold

Then average clustering linearly decreaseswith T from maximum at T = 0 to zero at T = 1Clustering is always zero at T > 1The model becomes identical to PA at T

Effective preferential attachmentPSO PAPSO PA S, wherePSO is popularity similarity optimizationPA is preferential attachment (popularity)S is similarity (sphere)PA is 1-dimensional (radial popularity)PSO is d1-dimensional, where d is the dimensionality of the similarity space

ValidationTake a series of historical snapshots of a real networkInfer angular/similarity coordinates for each nodeTest if the probability of new connections follows the model theoretical predictionLearning similarity coordinatesTake a historical snapshot of a real networkApply a maximum-likelihood estimation method (e.g., MCMC) using the static hyperbolic modelMetropolis-Hastings exampleAssign random coordinates to all nodesCompute current likelihoodSelect a random nodeMove it to a new random angular coordinateCompute new likelihood LnIf Ln > Lc, accept the moveIf not, accept it with probability Ln / LcRepeat

Real networksPGP web of trustNodes: PGP certificates (roughly, people)Links: Trust relationshipsInternetNodes: Autonomous systems (ASes)Links: Business relationshipsMetabolic (E.coli)Nodes: MetabolitesLinks: Reactions

Binning and overfittingMore rigorous measuresof modeling qualityNormalized likelihoodThe popularity similarity model does not describe well the actor network because very dissimilar actors often collaborate on big movies

Soft community detection effectInferred coordinates correlate with meaningful node groups

Capturing network structureAs a simple consequence of the fact the PSO model accurately describes the large-scale network growth dynamics, it also reproduces very well the observed large-scale network structure across a wide range of structural properties

Take-home messages (on PSO)Popularity similarity optimization dynamics Geometrical hyperbolicity Topological heterogeneity transitivity (real nets)Popularity is modeled by radial coordinatesSimilarity is modeled by angular coordinatesProjections of a properly weighted combination of all the factors shaping the network structureImmediate applications(submitted)New simple network-embedding methodThe idea is to replay the growth of a given network snapshot according to PSONew link prediction method, outperforming all the most popular link prediction methodsSome classes of links can be predicted with 100% accuracyPerhaps because the method captures all the factors shaping the network structureSomething is definitely wrongPlausible solutionGeometry under real networks is not hyperbolic but LorentzianLorentzian manifolds explicitly model timeProof that PSO graphs are random geometric graphs on de Sitter spacetime (accepted)Lorentzian manifoldsCausal structure

Alexandrov setsForm a base of the manifold topologySimilar to open balls in Riemannian caseRandom geometric graphDiscretization of a smooth manifolds (B.Riemann, Nature, v.7)

Take a circle of radius R

Sprinkle N points into it uniformly at random

Connect each pair of points iff the distance between them is x r RLorentzianLorentzian0; because Alexandrovsets are balls now

Major challenge (in progress)On the one hand, random Lorentzian graphs are random geometric graphs, and consequently exponential random graphs (equilibrium ensembles)On the other hand, they are dynamic growing graphs (non-equilibrium ensembles)Can it be the case that a given ensemble of graphs is static (equilibrium) and dynamic (non-equilibrium)at the same time???If we prove that it is indeed the case, then weDiscover some unseen static-dynamic graph dualityOpen a possibility to apply very powerful tools developed for equilibrium systems (e.g., exponential random graphs), to dynamic networksF. Papadopoulos, M. Kitsak, M. . Serrano, M. Bogu, and D. Krioukov,Popularity versus Similarity in Growing Networks,Nature, v.489, p.537, 2012