Critical window for connectivity in the Configuration Model

TL;DR

This paper determines the probability of connectivity in the configuration model within its critical window, based on degree distribution, and derives asymptotics for simple connected graphs with given degrees.

Contribution

It identifies the asymptotic connectivity probability in the critical window of the configuration model and derives related asymptotics for simple graphs with prescribed degrees.

Findings

Connectivity probability converges to a non-trivial limit.

Size of the non-giant component converges to a finite random variable.

Asymptotics for simple connected graphs with given degree sequences are established.

Abstract

We identify the asymptotic probability of a configuration model $\mathrm{CM}_n(\boldsymbol{d})$ to produce 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 the asymptotic number of simple connected graphs with a prescribed degree sequence follows.