Clustering means geometry in sparse graphs. Dmitri Krioukov Northeastern University Workshop on Big Graphs UCSD, San Diego, CA, January 2016

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1 in sparse graphs Dmitri Krioukov Northeastern University Workshop on Big Graphs UCSD, San Diego, CA, January 206

2 Motivation Latent space models Successfully used in sociology since the 70ies Recently shown to reproduce many structural properties of large real networks (with many applications) Which are necessary but not sufficient conditions for realism Pregeometric quantum gravity Define a random graph ensemble in purely graphstructural terms such that geometry is emergent, i.e., the ensemble is equivalent to a random geometric graph ensemble Motivation 2

3 Real networks Sparse Strong clustering Power laws Community structure Let s put the last two properties aside Background 3

4 Two properties to model Fixed number of edges Fixed number of triangles Plus we want to define an unbiased ensemble, graphs are random in all other respects Background 4

5 Strauss model Maximum-entropy random graphs with fixed number of nodes, edges, and triangles Exponential random graphs with P G = e H(G) /Z, H G = λ k k + λ t t, k = 2 t = 6 i,j a ij, i,j,k a ij a jk a ki. Background 5

6 Clustering In real networks Triangles are spread equally across all nodes In the Strauss model Triangles coalesce into a large clique The graphon is bipodal (Kenyon, Radin, Sadun) Strauss model has many other pathological properties (degeneracies, hysteresis, etc.) that are not observed in real networks Background 6

7 Clustering in the brain Background 7

8 Strauss can t be fixed for spreading H G = i (λ k,i k i + λ t,i t i ), s.t. λ k,i = λ k and λ t,i = λ t, where k i = j a ij and t i = a ij a jk a ki 2 j,k H G = λ k i k i + λ t i t i = 2λ k k + 3λ t t Background 8

9 Edge-independent random graphs Maximum-entropy random graphs with fixed expected values of edges (adjacency matrix elements, independent Bernoulli random variables) a ij = p ij H G = 2 i,j λ ij a ij p ij = / + e λ ij Z = + e λ ij i,j Problem formulation 9

10 Spreading problem is fixed k i = p ij j = k t i = p 2 j,k ij p jk p ki = t But we want an unbiased ensemble! The problem can be formulated using graphon entropy, after the following graphon modifications Problem formulation 0

11 Graphon modifications Dense graphons (intuitively, p ij p x, y ) If p 0, 2 [0,], then k x = n p x, y dy 0 and graphs are dense One way to get sparse graphs (used in L p graphons) is p n x, y = p(x, y)/n n k x t x 0 = n p n x, y dy 0 = n = p x, y dy, but p n x, y p n y, z p n z, x dy dz p x, y p y, z p z, x dy dz = -- clustering is zero Problem formulation

12 Graphon modifications Another way to sparsity is a la Caron-Fox p n n, n 2 0, 2 2 p n R 2 [0,] In the large-graph n limit, we have then k x = p x, y dy R t x = p x, y p y, z p z, x dy dz 2 R 2 Which can be both finite and positive Problem formulation 2

13 Graphon entropy density For a fixed n, the entropy density (entropy per node) of an exponential random graph ensemble with p ij = p(x i, y j ), where x i, y j n 2, n 2, is σ n p ij = H(p 2n i,j ij ), where H p = p log p p log p σ n p ij σ p = lim n 2n n 2,n 2 2 H p x, y dx dy (Janson-Hatami) Problem formulation 3

14 Problem formulation Find p x, y, a graphon defining the maximumentropy edge-independent random graphs with expected numbers of edges and triangles of all nodes fixed to values k, t Graphon p x, y maximizes σ p, constrained by k p = p x, y dy R = k t p = p x, y p y, z p z, x dy dz 2 R 2 = t Since the constraints do not depend on x, σ p = 2 R H p x, y dy Problem formulation 4

15 Problem solution Define Lagrangian L p = σ p + λ k k p + λ t t p L p p(x,y) = 0 yields the following integral equation log + 2λ p x,y k + λ t p y, z p z, x dz = 0 R which appears intractable We show that p x, y = +e β(ε μ), where ε = x y, β = λ t, and μ = 2λ k /λ t is an approximate solution Problem solution 5

16 Proof Define α = μβ, r = ε/2μ, a = e α, b = e 2αr log p x,y = β ε μ = 2α r 2 I r, α = 2μ R p y, z p z, x dz = α(b )(b a 2 ) [αr a2 b + a 2 2b 2 a 2 b + b 2a 2 log( + a) + b 2 a 2 log(a + b)] J r = r Proof 6

17 Proof I r, α J(r) /α for α Proof 7

18 Proof The expected number of common neighbors between x and y, p y, z p z, x dz = R 2μI r, α 2μJ r = 2μ r The red equation simplifies to α r 2 + λ k + λ t μ r = 0 With α = 2λ k and β = λ t yielding the solution Proof 8

19 Large-α asymptotics k 2μ + α e α -- expected # of edges t 3 2 μ2 π 3α c = 2t k π 3α σ β π expected # of triangles 2 -- expected clustering 3 αe α -- entropy density Problem solution 9

20 Summary Since entropy-maximizing graphon p x, y = p x y, these results mean that the ensemble of edge-independent random graphs with fixed expected degrees and clustering of all nodes, defined in purely graph-structural terms, is equivalent to the ensemble of soft random geometric graphs on the real line with the Fermi-Dirac connection probability p ε = /[ + exp β(ε μ) ] Surprisingly, this graph-structural characterization of a potentially rich geometric ensemble turned out to be rather simple, as it deals only with edges and triangles, evenly spread across all nodes Summary 20

21 Summary Caveats No proof of uniqueness The solution is approximate Becomes exact in the α = βμ limit In particular, in the standard (sharp) random geometric graphs with any μ > 0 and T = /β 0 (p x, y = Θ μ x y, k = 2μ, t = 3μ 2 /2, c = 3/4, σ = 0) Modulo these caveats, this appears to be the first characterization of a random geometric graph ensemble, purely in terms of graph-structural properties, plus the maximum-entropy principle Summary 2

arxiv: v3 [math.pr] 10 Oct 2017

arxiv: v3 [math.pr] 10 Oct 2017 Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution Pim van der Hoorn 1, Gabor Lippner 2, and Dmitri Krioukov 1,2,3 arxiv:175.1261v3 [math.pr] 1 Oct 217 1 Northeastern University,