Efficient Computation of Maximum Flexi-Clique in Networks

2026-02-11

AI Summary

This work addresses the limitations of traditional dense subgraph models, which rely on fixed density thresholds and struggle to balance connectivity and cohesiveness in large-scale subgraphs. To overcome this, we propose the Flexi-clique model, whose degree constraints grow sublinearly with subgraph size, thereby transcending rigid density requirements. We develop a fast heuristic algorithm (FPA) and an exact branch-and-bound algorithm (EBA), integrating core decomposition–based seed generation, connectivity-aware pruning, and multi-rule enhancement strategies to effectively trade off solution quality and computational efficiency. Experimental results demonstrate that FPA achieves near-optimal solutions with low overhead, while EBA efficiently computes exact solutions, significantly improving the scalability and practicality of cohesive subgraph discovery on both real-world and synthetic large graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Heuristic Search
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Discovering large cohesive subgraphs is a key task for graph mining. Existing models, such as clique, k-plex, and {\gamma}-quasi-clique, use fixed density thresholds that overlook the natural decay of connectivity as the subgraph size increases. The Flexi-clique model overcomes this limitation by imposing a degree constraint that grows sub-linearly with subgraph size. We provide the algorithmic study of Flexi-clique, proving its NP-hardness and analysing its non-hereditary properties. To address its computational challenge, we propose the Flexi-Prune Algorithm FPA, a fast heuristic using core-based seeding and connectivity-aware pruning, and the Efficient Branch-and-Bound Algorithm EBA, an exact framework enhanced with multiple pruning rules. Experiments on large real-world and synthetic networks demonstrate that FPA achieves near-optimal quality at much lower cost, while EBA efficiently computes exact solutions. Flexi-clique thus provides a practical and scalable model for discovering large, meaningful subgraphs in complex networks.

Problem

• Flexi-clique
• cohesive subgraph
• density threshold
• graph mining
• NP-hardness

Innovation

• Flexi-clique
• subgraph mining
• NP-hardness
• branch-and-bound
• core-based pruning