The distribution of clusters in random graphs

Abstract

AbstractGiven a random graph, we investigate the occurrence of subgraphs especially rich in edges. Specifically, given a ϵ [0,1], a set of k points in a graph G is defined to be an a-cluster of cardinality k if the induced subgraph contains at least ak2 edges, so that in the extreme case a = 1, an a-cluster is the same as a clique. We let G = G(n, p) be a random graph on n vertices with edges chosen independently with probability p. Let W denote the number of a-clusters of cardinality k in G, where k and n tend to infinity so that the expected number λ of a-clusters of cardinality k does not grow or decay too rapidly. We prove that W is asymptotically distributed as Zλ, whose distribution is Poisson with mean λ, which is the same result that Bollobás and Erdös have proved for cliques. In contrast to the situation for cliques (a = 1) however, for all a < 1 the second moment of W blows up, i.e., the expected number of neighbors of a given cluster tends to infinity. Nevertheless, the probability that there exists at least one pair of neighboring clusters tends to zero, and a Poisson approximation for W is valid