Universality for the Distance in Finite Variance Random Graphs

We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the classical Erdős-Rényi graph). In the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes. We consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two, so that, in particular, the degrees have finite variance. We prove that the graph distance grows like log ν N, where the ν depends on the capacities and N denotes the size of the graph. In addition, the random fluctuations around this asymptotic mean log ν N are shown to be tight. We also consider the case where the capacities are independent copies of a positive random Λ with $\mathbb{P}\left (\Lambda>x\right )\leq cx^{1-\tau}$ , for some constant c and τ>3, again resulting in graphs where the degrees have finite variance. The method of proof of these results is to couple each member of the class to the Poissonian random graph, for which we then give the complete proof by adapting the arguments of van der Hofstad et al. (Random Struct. Algorithms 27(2):76–123, 2005).