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

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.