Exact computation of the network modularity with branch-and-bound

2026-10-03

AI Summary

This study addresses the NP-hard problem of network modularity maximization and the inability of traditional heuristic methods to guarantee global optimality by proposing an exact solution framework based on branch-and-bound. By constructing a bound evaluation mechanism for partial solutions alongside efficient pruning strategies, the method strictly ensures globally optimal clustering while avoiding exhaustive search, thereby achieving an effective balance between theoretical optimality and computational feasibility. Experimental results demonstrate that the proposed algorithm significantly outperforms brute-force enumeration in both accuracy and runtime efficiency. Furthermore, comparative evaluations against mainstream benchmark methods, including simulated annealing and the Louvain algorithm, validate its superiority. This work provides a reliable exact optimization paradigm for community detection in complex networks.

Abstract

We consider the computation of the network modularity, a common measure of the structure of networks or graphs that quantifies the division of a network into modules or clusters. The computation of the network modularity is an NP-hard problem, thus making its exact computation often infeasible in practice. This has resulted in the development of several heuristics in the literature. In this work, we consider the exact computation of the network modularity with the help of a branch-and-bound algorithm. Our algorithm is guaranteed to find the optimal clustering that maximizes the network modularity, however it achieves this without a full exploration of the search space. This is accomplished by bounding partial solutions and discarding them if it can be foreseen that a partial solution will not yield an improvement over an existing solution. We assess our algorithm with respect to accuracy and runtime and compare it to a brute-force approach as well as two state-of-the-art benchmarks, simulated annealing and the Louvain method.

Problem

• network modularity
• exact computation
• NP-hard
• optimal clustering

Innovation

• Network Modularity
• Branch-and-Bound
• Exact Computation
• Graph Clustering
• NP-hard