Asymptotics for the size of the largest component scaled to “logn” in inhomogeneous random graphs

Abstract

We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by log n, with n being the size of the graph. This generalizes a result for the “rank-1 case”. We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the negative solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.