Large cliques in a power-law random graph

Abstract

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

13 pages