Large cliques in a power-law random graph

Abstract

We study the size of the largest clique $ω(G(n,α))$ in a random graph $G(n,α)$ on $n$ vertices which has power-law degree distribution with exponent $α$. We show that for `flat' degree sequences with $α>2$ whp the largest clique in $G(n,α)$ is of a constant size, while for the heavy tail distribution, when $0<α<2$, $ω(G(n,α))$ grows as a power of $n$. Moreover, we show that a natural simple algorithm whp finds in $G(n,α)$ a large clique of size $(1+o(1))ω(G(n,α))$ in polynomial time.