Small clique counts in dense inhomogeneous random graphs

Abstract

Given a symmetric measurable function (graphon) W : [0, 1]2 → [0, 1] the random graph G(n, W) on vertex set [n] = {1, . . . , n} is defined as follows. Sample uniform independent points x1, . . . , xn ∈ [0, 1]. For every pair of vertices i and j connect them independently with probability W(xi , xj ). We study a specific summary statistic of this random graph, namely the number X of cliques of fixed size. We show that, depending on W, the asymptotic fluctuations of X around the expectation are either normal or a sum of squares of independent normal variables.