Asymptotics for the size of the largest component scaled to "log n" in inhomogeneous random graphs

TL;DR

This paper derives an exact formula for the scaled size of the largest component in inhomogeneous random graphs in the subcritical case, revealing the role of survival probability equations similar to the supercritical case.

Contribution

It generalizes previous results to the subcritical case, showing that the same survival probability equations determine the largest component size.

Findings

Exact formula for largest component size scaled to log n

Survival probability equations are crucial in subcritical analysis

Generalization of rank 1 case results

Abstract

We study the inhomogeneous random graphs in the subcritical case. We derive an exact formula for the size of the largest connected component scaled to $\log n$ where $n$ is the size of the graph. This generalizes the recent result for the "rank 1 case". Here 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, plays the crucial role in the subcritical case as well. But now these are the negative solutions which come into play.