B. G. Pittel

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

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 $n^{1/γ+\varepsilon_n}$, $\varepsilon_n=O(\ln\ln n/\ln n)$, $1/γ$ 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.