Large cliques in a power-law random graph

TL;DR

This paper analyzes the size of the largest clique in power-law random graphs, revealing phase transitions based on degree distribution and proposing an efficient algorithm to find large cliques.

Contribution

It characterizes the largest clique size in power-law graphs for different degree exponents and introduces a polynomial-time algorithm to find near-maximum cliques.

Findings

Largest clique size is constant for $\alpha>2$

Clique size grows as a power of n for $0<\alpha<2$

Simple algorithm finds large cliques with high probability

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.