Aggregating maximal cliques in real-world graphs

TL;DR

This paper introduces  ho-dense aggregators, a new method for summarizing maximal clique structures in graphs efficiently, especially in real-world networks, by capturing key clusters instead of enumerating all cliques.

Contribution

The paper proposes  ho-dense aggregators, providing theoretical bounds and algorithms for concise, near-linear time summaries of maximal clique structures in graphs.

Findings

Algorithms achieve sub-exponential size aggregators for all  ho<1

Near-linear time algorithms for graphs with bounded degeneracy

Empirical results show significant speedups and more concise summaries

Abstract

Maximal clique enumeration is a fundamental graph mining task, but its utility is often limited by computational intractability and highly redundant output. To address these challenges, we introduce \emph{$\rho$-dense aggregators}, a novel approach that succinctly captures maximal clique structure. Instead of listing all cliques, we identify a small collection of clusters with edge density at least $\rho$ that collectively contain every maximal clique. In contrast to maximal clique enumeration, we prove that for all $\rho < 1$, every graph admits a $\rho$-dense aggregator of \emph{sub-exponential} size, $n^{O(\log_{1/\rho}n)}$, and provide an algorithm achieving this bound. For graphs with bounded degeneracy, a typical characteristic of real-world networks, our algorithm runs in near-linear time and produces near-linear size aggregators. We also establish a matching lower bound on aggregator size, proving our results are essentially tight. In an empirical evaluation on real-world networks, we demonstrate significant practical benefits for the use of aggregators: our algorithm is consistently faster than the state-of-the-art clique enumeration algorithm, with median speedups over $6\times$ for $\rho=0.1$ (and over $300\times$ in an extreme case), while delivering a much more concise structural summary.