Distances in random graphs with finite variance degrees

In this paper we study a random graph with N nodes, where node j has degree Dj and{Dj}j=1 are i.i.d. with P(Dj ≤ x) = F (x). We assume that 1 − F (x) ≤ cx for some τ > 3and some constant c > 0. This graph model is a variant of the so-called configuration model,and includes heavy tail degrees with finite variance.The minimal number of edges between two arbitrary connected nodes, also known as thegraph distance or the hopcount, is investigated when N → ∞. We prove that the graph distancegrows like logν N , when the base of the logarithm equals ν = E[Dj(Dj − 1)]/E[Dj] > 1. Thisconfirms the heuristic argument of Newman, Strogatz and Watts [35]. In addition, the randomfluctuations around this asymptotic mean logν N are characterized and shown to be uniformlybounded. In particular, we show convergence in distribution of the centered graph distance alongexponentially growing subsequences.