Critical Window for Connectivity in the Configuration Model

Abstract

We identify the asymptotic probability of a configuration model CMn(d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.