Finding Community Structure with Performance Guarantees in Complex Networks

TL;DR

This paper introduces two polynomial-time algorithms for community detection in networks with performance guarantees, one providing a constant-factor approximation for power-law networks and the other offering a scalable LP-based method with high accuracy.

Contribution

It presents the first approximation algorithms with theoretical guarantees for modularity maximization in community detection, including a scalable LP-based approach.

Findings

The first algorithm guarantees modularity within a constant factor for power-law networks.

The second algorithm is a fast, scalable LP-based method with high solution accuracy.

Experiments show the rounding algorithm often finds optimal solutions and scales to larger networks.

Abstract

Many networks including social networks, computer networks, and biological networks are found to divide naturally into communities of densely connected individuals. Finding community structure is one of fundamental problems in network science. Since Newman's suggestion of using \emph{modularity} as a measure to qualify the goodness of community structures, many efficient methods to maximize modularity have been proposed but without a guarantee of optimality. In this paper, we propose two polynomial-time algorithms to the modularity maximization problem with theoretical performance guarantees. The first algorithm comes with a \emph{priori guarantee} that the modularity of found community structure is within a constant factor of the optimal modularity when the network has the power-law degree distribution. Despite being mainly of theoretical interest, to our best knowledge, this is the first approximation algorithm for finding community structure in networks. In our second algorithm, we propose a \emph{sparse metric}, a substantially faster linear programming method for maximizing modularity and apply a rounding technique based on this sparse metric with a \emph{posteriori approximation guarantee}. Our experiments show that the rounding algorithm returns the optimal solutions in most cases and are very scalable, that is, it can run on a network of a few thousand nodes whereas the LP solution in the literature only ran on a network of at most 235 nodes.