Identifying network communities with a high resolution

Abstract

Community structure is an important property of complex networks. The automatic discovery of such structure is a fundamental task in many disciplines, including sociology, biology, engineering, and computer science. Recently, several community discovery algorithms have been proposed based on the optimization of a modularity function (Q) . However, the problem of modularity optimization is NP-hard and the existing approaches often suffer from a prohibitively long running time or poor quality. Furthermore, it has been recently pointed out that algorithms based on optimizing Q will have a resolution limit; i.e., communities below a certain scale may not be detected. In this research, we first propose an efficient heuristic algorithm QCUT, which combines spectral graph partitioning and local search to optimize Q . Using both synthetic and real networks, we show that QCUT can find higher modularities and is more scalable than the existing algorithms. Furthermore, using QCUT as an essential component, we propose a recursive algorithm HQCUT to solve the resolution limit problem. We show that HQCUT can successfully detect communities at a much finer scale or with a higher accuracy than the existing algorithms. We also discuss two possible reasons that can cause the resolution limit problem and provide a method to distinguish them. Finally, we apply QCUT and HQCUT to study a protein-protein interaction network and show that the combination of the two algorithms can reveal interesting biological results that may be otherwise undetected.

Edge ratio and community structure in networks

“…We reproduced these results with our version of the Grötschel and Wakabayashi algorithm. The optimal partition for modularity maximization contains the following 5 communities: C m 1 = {1, 2, 3,5,6,7,8,19,29, 30} with 6 n and 4 c, C m 2 = {4, 9,10,11,12,13,14,15,16,17,18,20,21,22,23,24,25,26,27,28,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,54,55,56 and 3 c. These 5 communities consist of two large ones with no (for c) or very few (for l) misclassifications, two small communities with both n and c books and one community with all three categories. We count misclassifications as follows: any l in a community with a majority of c's or n's or conversely counts for 1;…”

“…A large number of heuristics were proposed to maximize modularity. They rely on simulated annealing [16], extremal optimization [17], mean field annealing [18], genetic search [19], dynamical clustering [20], multilevel partitioning [21], contraction-dilation [22], multistep greedy [23], quantum mechanics [24] and a variety of other approaches [25,26,27,28,29,30]. These heuristics provide, usually in moderate time, near optimal partitions for the modularity criterion or, possibly, optimal partitions but without the proof of their optimality.…”

Improving heuristics for network modularity maximization using an exact algorithm

“…Many heuristics have been proposed, while exact algorithms for modularity maximization are rare. Heuristics are based on agglomerative hierarchical clustering [2][3][4][5][6], simulated annealing [7][8][9], mean field annealing [10], genetic search [11], extremal optimization [12], spectral clustering [13], linear programming followed by randomized rounding [14], dynamical clustering [15], multilevel partitioning [16], contraction-dilation [17], multistep greedy search [18], quantum mechanics [19] and many more [6,[20][21][22][23][24]. These heuristics are able to solve large instances with up to hundred or thousand entities and therefore are often preferred to exact algorithms, even though they do not have a guarantee of optimality.…”

Locally optimal heuristic for modularity maximization of networks

“…The partitioning and hybrid heuristics rely upon simulated annealing [24,34,35], mean field annealing [32], genetic search [51], extremal optimization [16], linear programming followed by randomized rounding [1], dynamical clustering [6], multilevel partitioning [15], contraction-dilation [36], multistep greedy search [49], quantum mechanics [44] and many more sources of inspiration [10,50,48,17,31].…”