The size of the largest component below phase transition in inhomogeneous random graphs

TL;DR

This paper derives an exact formula for the asymptotic size of the largest component in inhomogeneous random graphs below the phase transition, completing the understanding of component sizes in both subcritical and supercritical regimes.

Contribution

It introduces a new formula for the largest component size in the subcritical phase of inhomogeneous random graphs, extending previous results to a more complete phase transition analysis.

Findings

Exact asymptotic formula for largest component size in subcritical case

Completes the phase transition analysis by matching supercritical results

Provides applications of the new formula in specific examples

Abstract

We study the "rank 1 case" of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result is new, it completes the corresponding known result in the supercritical case. We provide some examples of application of a new formula.