On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

Abstract

A uniformly random graph on n vertices with a fixed degree sequence, obeying a γ subpower law, is studied. It is shown that, for γ>3, in a subcritical phase with high probability the largest component size does not exceed n^1/γ+ɛ_n, ɛ_n=O(ln ln n/ln n), 1/γ being the best power for this random graph. This is similar to the best possible n^1/(γ−1) bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.