Maximal Cliques in Scale-Free Random Graphs

TL;DR

This paper analyzes the number of maximal cliques in various random graph models, revealing super-polynomial lower bounds in some cases and explaining discrepancies with empirical observations through experiments.

Contribution

It provides new theoretical bounds on maximal cliques in different network models and reconciles these with empirical data via experimental analysis.

Findings

Super-polynomial lower bounds for inhomogeneous random graphs.

Linear and polynomial upper bounds for sparse Erdős-Rényi graphs.

Experiments show asymptotic behavior dominates only in extremely large networks.

Abstract

We investigate the number of maximal cliques, i.e., cliques that are not contained in any larger clique, in three network models: Erd\H{o}s-R\'enyi random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and geometric inhomogeneous random graphs. For sparse and not-too-dense Erd\H{o}s-R\'enyi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (geometric) inhomogeneous random graphs are sparse, we give super-polynomial lower bounds for these models. This comes from the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (geometric) inhomogeneous random graphs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size.