Aggregating maximal cliques in real-world graphs

2025-12-03

AI Summary

Maximal clique enumeration in real-world graphs suffers from high computational complexity and severe output redundancy, limiting practical applicability. Method: We propose the ρ-dense aggregator—a novel, compact structure that covers all maximal cliques using a small number of high-density subgraphs. Algorithmically, we integrate structural analysis of maximal cliques with graph degeneracy, designing density-constrained clustering and efficient subgraph search. Contribution/Results: Theoretically, we prove a subexponential upper bound on aggregator size for arbitrary graphs and establish a near-linear-time construction algorithm for bounded-degeneracy graphs, accompanied by a matching lower bound—demonstrating tightness. Experimentally, on real-world networks, our method achieves median speedups exceeding 6× (ρ = 0.1), with extreme cases surpassing 300×, while drastically compressing output size. This enables efficient, scalable summarization of clique structures.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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{$ ho$-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 $ ho$ that collectively contain every maximal clique. In contrast to maximal clique enumeration, we prove that for all $ ho<1$, every graph admits a $ ho$-dense aggregator of emph{sub-exponential} size, $n^{O(log_{1/ ho}n)}$, 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 $6 imes$ for $ ho=0.1$ (and over $300 imes$ in an extreme case), while delivering a much more concise structural summary.

Problem

• Summarizes maximal cliques via dense clusters to reduce redundancy.
• Achieves sub-exponential aggregator size for computational efficiency.
• Enables near-linear time aggregation in real-world degenerate graphs.

Innovation

• Clusters with edge density replace exhaustive clique enumeration
• Sub-exponential size aggregators for all graphs with proven bounds
• Near-linear time algorithm for real-world networks with degeneracy