Asymptotic normality of global clustering coefficient of uniform random intersection graph

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1 Asymptotic normality of global clustering coefficient of uniform random intersection graph Mindaugas Bloznelis joint work with Jerzy Jaworski Vilnius University, Lithuania bloznelis May 16, 2018

2 Global clustering coefficient The global clustering coefficient of a finite graph G is the ratio C G = 3N /N, N is the number of triangles, N is the number of 2 paths C G represents the probability that a randomly selected path of length 2 induces triangle in G C G is a commonly used network characteristic, assessing the strength of the statistical association between neighboring adjacency relations For example, in a social network the tendency of linking actors which have a common neighbor is reflected by a non-negligible value of the global clustering coefficient

3 Random intersection graph, RIG Clustering in a social network can be explained by an auxiliary bipartite structure: each actor is prescribed a collection of attributes and any two actors sharing a common attribute have high chances of being adjacent, cf [Newman et all Random graphs with arbitrary degree distributions and their applications PhysRevE, 2002] The respective random intersection graph (RIG on the vertex set V = {v 1,, v n } and with the auxiliary attribute set W = {w 1,, w m } defines adjacency relations with the help of a random bipartite graph H linking actors (=vertices to attributes: two actors are adjacent in RIG if they have a common neighbour in H For H drawn uniformly at random from the class of bipartite graphs where each actor v i V has exactly r neighbours in W we obtain the uniform RIG denoted G = G(n, m, r Alternative definition: vertices v 1,, v n of G are represented by iid random subsets S 1,, S n W, each of size r Vertices v i, v j are adjacent whenever S i S j

4 Uniform random intersection graph G(n, m, r Vertices v 1,, v n of G = G(n, m, r are represented by iid random subsets S 1,, S n W = {w 1,, w m }, each of size r Vertices v i, v j are adjacent whenever S i S j The graph has been widely studied in the literature mainly as a model of secure wireless sensor network that uses random predistribution of keys [Eschenauer and Gligor (2002]: each sensor v i is prescribed a collection S i of secret keys; two sensors establish a secure communication link if they share a secret key We consider large random intersection graphs, where m, n For r 2 = o(m the edge probability is p e = P(v i v j = r 2 m 1 + O(r 4 m 2 We are interested in sparse graphs with bounded average degree η := (n 1p e (n 1r 2 m 1 Paveiksliukas!

5 Asymptotic normality of C G = 3N /N (I Let G be an instance of G(n, m, r Denote µ = 3EN EN, N = N EN EN, N = N EN EN We approximate C G by µ The approximation error ( N C G µ = µ + 1 N = µ(n N R (1 has the leading term µ(n N and (negligible remainder N R = µ(n N N + 1 Assume ( that EN, EN + and properly standardized vector N, N is asymptotically (bivariate normal Then (1 implies the asymptotic normality of C G µ

6 Asymptotic normality of C G = 3N /N (I Let G be an instance of G(n, m, r Denote µ = 3EN EN, N = N EN EN, N = N EN EN We approximate C G by µ The approximation error ( N C G µ = µ + 1 N = µ(n N R (1 has the leading term µ(n N and (negligible remainder N R = µ(n N N + 1 Assume ( that EN, EN + and properly standardized vector N, N is asymptotically (bivariate normal Then (1 implies the asymptotic normality of C G µ Remark We have µ = P(v 2 v 3 v 1 v 2, v 1 v 3 = P ( v2 v 3 v1 v 2, v 1 v 3, where the vertex triple (v1, v 2, v 3 is drawn at random

7 Bivariate asymptotic normality of ( N, N Denote σ 2 = VarN, σ 2 = VarN, σ = Cov(N, N Proposition Let m, n + Assume that r 2 and r = O(1 Assume that σ 2 +, σ2 + Suppose that the ratio σ /(σ σ converges to a limit We denote the limit ρ Then the random vector ( σ 1 (N EN, σ 1 (N EN converges in distribution to a Gaussian random vector (ξ, ξ, where Eξ j = 0, Eξj 2 = 1, j =,, and Eξ ξ = ρ Remark about the scaling of σ and σ We have for r 3 = o(m σ 2 1 2r 2 nη r nη2, σ 2 nη 3( r + 1 2r 2 + nη 2( , r σ nη 3( 1 2r nη 2 1 r 2r Recall that η = (n 1p e nr 2 m 1 denotes the average degree For fixed r and bounded η η 0 > 0 we have ρ η 0 (1 + 2r + r (η r(η 0 (4r 2 + 4r r 2 + 2r

8 Asymptotic normality of C G = 3N /N (II The bivariate asymptotic normality (Proposition ( N EN σ 1, N EN σ 1 (ξ, ξ implies the approximation (in distribution ( N EN C G µ N EN ( σ µ ξ σ ξ ( EN EN EN EN Hence the asymptotic normality of C G Comment on the contribution of N and N to (*: for r = O(1 the coefficients σ /EN and σ /EN are of the same order of magnitude, thus, N and N enter on equal terms; for r +, η = o(r we have σ /EN = o ( σ /EN, that is, the triangle count N outplays N These observations follow easily from the relations Varξ = Varξ = 1, µ = 3EN /EN r 1, σ 2 1 2r 2 nη r nη2, σ 2 nη 3( 2+ 2 r + 1 2r 2 +nη 2( r 2 1

9 Approach to the asymptotic normality of ( N, N It suffices to show that for any reals a, b the random variables X n,m = a N EN σ + b N EN σ converge in distribution to aξ + bξ as n, m + To this aim we use Stein s method In our case this approach leads to awkward calculations To make calculations feasible we preprocess the subgraph counts N = N (S 1,, S n = i,j,k, (1 N = N (S 1,, S n = {i,j,k} [n] {i,j,k} [n] ( ijk + jki + kij Namely, we decompose them into series of uncorrelated polynomials of increasing orded, called Hoeffding s decomposition The decomposition helps to tackle statistical dependencies In particular, we obtain reasonably simple expressions of Var(N, Var(N and Cov(N, N