Toward Optimal Community Detection: From Trees to General Weighted Networks

Abstract Many networks, including the Internet, social networks, and biological relations, are found to be naturally divided into communities of densely connected nodes, known as community structure. Since Newman’s suggestion of using modularity as a measure to qualify the goodness of community structures, many efficient methods to maximize modularity have been proposed but without optimality guarantees. In this work we study exact and theoretically near-optimal algorithms for maximizing modularity. In the first part, we investigate the complexity and approximability of the problem on tree graphs. Somewhat surprisingly, the problem is still NP-complete on trees. We then provide a polynomial time algorithm for uniform-weighted trees and a pseudopolynomial time algorithm and a PTAS for trees with arbitrary weights. In the second part, we present a family of compact linear programming formulations for the problem in general graphs. These formulations exploit the graph connectivity structure and reduce substantially the number of constraints, thus, they vastly improve the running time for solving linear programming and integer programming. As a result, networks of thousands of vertices can be solved in minutes, whereas the current largest instance solved with mathematical programming has fewer than 250 vertices.