Susceptibility of random graphs with given vertex degrees

TL;DR

This paper analyzes the susceptibility, or mean cluster size, in random graphs with specified degrees, establishing convergence to branching process expectations and exploring critical behavior and exponents.

Contribution

It provides new theoretical results on the convergence of susceptibility in degree-constrained random graphs and examines critical phenomena and exponents.

Findings

Susceptibility converges to branching process expectations under weak assumptions.

In the supercritical case, a modified susceptibility ignoring the giant component is analyzed.

Critical exponents differ on subcritical and supercritical sides in certain examples.

Abstract

We study the susceptibility, i.e., the mean cluster size, in random graphs with given vertex degrees. We show, under weak assumptions, that the susceptibility converges to the expected cluster size in the corresponding branching process. In the supercritical case, a corresponding result holds for the modified susceptibility ignoring the giant component and the expected size of a finite cluster in the branching process; this is proved using a duality theorem. The critical behaviour is studied. Examples are given where the critical exponents differ on the subcritical and supercritical sides.