Upper bounds for the largest components in critical inhomogeneous random graphs

TL;DR

This paper analyzes the size of the largest components in the critical inhomogeneous Norros-Reittu random graph, providing improved upper bounds on the probability of large clusters and simplifying bounds for small components.

Contribution

It introduces stronger upper bounds for large component probabilities and simplifies the derivation of bounds for small components in the critical inhomogeneous random graph model.

Findings

Stronger upper bounds for the probability of large maximal clusters.

Simplified derivation of polynomial upper bounds for small largest components.

Enhanced understanding of component size distributions at criticality.

Abstract

We consider the Norros-Reittu random graph $NR_n(\textbf{w})$, where edges are present independently but edge probabilities are moderated by vertex weights, and use probabilistic arguments based on martingales to analyse the component sizes in this model when considered at criticality. In particular, we obtain stronger upper bounds (with respect to those available in the literature) for the probability of observing unusually large maximal clusters, and simplify the arguments needed to derive polynomial upper bounds for the probability of observing unusually small largest components.