Efficient Computation of Maximum Flexi-Clique in Networks

TL;DR

The paper tackles discovering large cohesive subgraphs by introducing Flexi-clique, a size-adaptive model with a sub-linear degree constraint $\delta(G[H]) \ge \lfloor |H|^{\tau} \rfloor$. It proves NP-hardness and non-hereditaryity, and offers two complementary algorithms: a fast heuristic (FPA) using core-based seeding and connectivity-aware pruning, and an exact branch-and-bound method (EBA) employing connectivity-preserving branching with six pruning rules. Experimental results on real and synthetic networks show that FPA delivers near-optimal quality at low cost, while EBA efficiently computes exact solutions, validating Flexi-clique as a scalable and meaningful model for large-network analysis. The approach provides practical tools for capturing size-dependent cohesion that aligns with real-world community evolution and density patterns.

Abstract

Discovering large cohesive subgraphs is a key task for graph mining. Existing models, such as clique, k-plex, and γ-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.

Figures (14)

• Figure 1: Relationships between subgraph size and degree
• Figure 2: Reduction process
• Figure 3: Non-hereditary property
• Figure 4: Observations on core decomposition
• Figure 5: FPA example $(\tau = 0.75)$

Theorems & Definitions (21)

• Definition 1: $k$-core seidman1983network
• Definition 2: Densest subgraph lanciano2024survey
• Definition 3: $k$-clique
• Definition 4: Flexi$-$clique flexi
• Theorem 1
• Theorem 2
• proof
• proof
• proof
• Example 1