On Finding the Community with Maximum Persistence Probability

TL;DR

This paper introduces a new mathematical programming approach and heuristic algorithms to efficiently identify communities with maximum persistence probability in networks, addressing computational challenges in community detection.

Contribution

It develops an integer fractional programming model for community detection based on persistence probability and proposes heuristic methods for practical solutions in larger networks.

Findings

Effective heuristic algorithms for community detection based on persistence probability.

Validated approach on simulated networks with real network comparisons.

Demonstrated the method's reliability and effectiveness in identifying communities.

Abstract

The persistence probability is a statistical index that has been proposed to detect one or more communities embedded in a network. Even though its definition is straightforward, e.g, the probability that a random walker remains in a group of nodes, it has been seldom applied possibly for the difficulty of developing an efficient algorithm to calculate it. Here, we propose a new mathematical programming model to find the community with the largest persistence probability. The model is integer fractional programming, but it can be reduced to mixed-integer linear programming with an appropriate variable substitution. Nevertheless, the problem can be solved in a reasonable time for networks of small size only, therefore we developed some heuristic procedures to approximate the optimal solution. First, we elaborated a randomized greedy-ascent method, taking advantage of a peculiar data structure to generate feasible solutions fast. After analyzing the greedy output and determining where the optimal solution is eventually located, we implemented improving procedures based on a local exchange, but applying different long term diversification principles, that are based on variable neighborhood search and random restart. Next, we applied the algorithms on simulated graphs that reproduce accurately the clustering characteristics found in real networks to determine the reliability and the effectiveness of our methodology. Finally, we applied our method to two real networks, comparing our findings to what found by two well-known alternative community detection procedures.