A Phase Transition for the Diameter of the Configuration Model

Abstract

In this paper, we study the configuration model (CM) with independent and identically-distributed (i.i.d.) degrees. We establish a phase transition for the diameter when the power-law exponent $\tau$ of the degrees satisfies $\tau \in (2,3)$. Indeed, we show that for $\tau>2$ and when vertices with degree $1$ or $2$ are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for $\tau \in (2,3)$, the diameter of the graph is, with high probability, bounded from above by a constant times the $\log \log$ of the size of the graph.