The largest component in a subcritical random graph with a power law degree distribution

TL;DR

This paper proves that in subcritical random graphs with power law degree distributions (exponent > 3), the largest component size scales with the maximum vertex degree, confirming a conjecture by Durrett.

Contribution

It establishes the size of the largest component in subcritical power law random graphs and extends results to other models, confirming Durrett's conjecture.

Findings

Largest component size scales as a constant times maximum degree

Results apply to multiple random graph models with power law distributions

Confirms a conjecture by Durrett

Abstract

It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent $\gamma>3$, the largest component is of order $n^{1/(\gamma-1)}$. More precisely, the order of the largest component is approximatively given by a simple constant times the largest vertex degree. These results are extended to several other random graph models with power law degree distributions. This proves a conjecture by Durrett.