Reformulation of a locally optimal heuristic for modularity maximization

National audienceA network, or graph, G = (V,E) consists of a set of vertices V = {1, . . . , n} and a set of edges E = {1, . . . ,m} connecting vertices. One of the most studied problems in the field of complex systems is to find communities, or clusters, in networks. A community consists of a subset S of the vertices of V where inner edges connecting pairs of vertices of S are more dense than cut edges connecting vertices of S to vertices of V \S. Many criteria have been proposed to evaluate partitions of V into communities

Additive Approximation Algorithms for Modularity Maximization

The modularity is a quality function in community detection, which was introduced by Newman and Girvan (2004). Community detection in graphs is now often conducted through modularity maximization: given an undirected graph $G=(V,E)$, we are asked to find a partition $\mathcal{C}$ of $V$ that maximizes the modularity. Although numerous algorithms have been developed to date, most of them have no theoretical approximation guarantee. Recently, to overcome this issue, the design of modularity maximization algorithms with provable approximation guarantees has attracted significant attention in the computer science community. In this study, we further investigate the approximability of modularity maximization. More specifically, we propose a polynomial-time $\left(\cos\left(\frac{3-\sqrt{5}}{4}\pi\right) - \frac{1+\sqrt{5}}{8}\right)$-additive approximation algorithm for the modularity maximization problem. Note here that $\cos\left(\frac{3-\sqrt{5}}{4}\pi\right) - \frac{1+\sqrt{5}}{8} < 0.42084$ holds. This improves the current best additive approximation error of $0.4672$, which was recently provided by Dinh, Li, and Thai (2015). Interestingly, our analysis also demonstrates that the proposed algorithm obtains a nearly-optimal solution for any instance with a very high modularity value. Moreover, we propose a polynomial-time $0.16598$-additive approximation algorithm for the maximum modularity cut problem. It should be noted that this is the first non-trivial approximability result for the problem. Finally, we demonstrate that our approximation algorithm can be extended to some related problems.Comment: 23 pages, 4 figure

Using mathematical programming to refine heuristic solutions for network clustering

International audienceWe propose mathematical programming based aproaches to refine graph clustering solutions computed by heuristics. Clustering partitions are refined by applying cluster splitting and a combination of merging and splitting actions. A refinement scheme based on iteratively fixing and releasing integer variables of a mixed-integer quadratic optimization formulation appears to be particularly efficient. Computational experiments show the effectiveness and efficiency of the proposed approaches

Column generation algorithms for exact modularity maximization in networks

International audienceFinding modules, or clusters, in networks currently attracts much attention in several domains. The most studied criterion for doing so, due to Newman and Girvan [Phys. Rev. E 69, 026113 (2004)], is modularity maximization. Many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality. There are few exact algorithms, prominent among which is a paper by Xu et al. [Eur. Phys. J. B 60, 231 (2007)]. Modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of Grötschel and Wakabayashi [Math. Program. 45, 59 (1989)] applied. We propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming. Performance of the four resulting algorithms is compared on problems from the literature. Instances with up to 512 entities are solved exactly. Moreover, the computing time of previously solved problems are reduced substantially

The Bayan Algorithm: Detecting Communities in Networks Through Exact and Approximate Optimization of Modularity

Community detection is a classic problem in network science with extensive applications in various fields. Among numerous approaches, the most common method is modularity maximization. Despite their design philosophy and wide adoption, heuristic modularity maximization algorithms rarely return an optimal partition or anything similar. We propose a specialized algorithm, Bayan, which returns partitions with a guarantee of either optimality or proximity to an optimal partition. At the core of the Bayan algorithm is a branch-and-cut scheme that solves an integer programming formulation of the problem to optimality or approximate it within a factor. We demonstrate Bayan's distinctive accuracy and stability over 21 other algorithms in retrieving ground-truth communities in synthetic benchmarks and node labels in real networks. Bayan is several times faster than open-source and commercial solvers for modularity maximization making it capable of finding optimal partitions for instances that cannot be optimized by any other existing method. Overall, our assessments point to Bayan as a suitable choice for exact maximization of modularity in networks with up to 3000 edges (in their largest connected component) and approximating maximum modularity in larger networks on ordinary computers.Comment: 6 pages, 2 figures, 1 tabl

COMMUNITY DETECTION IN COMPLEX NETWORKS AND APPLICATION TO DENSE WIRELESS SENSOR NETWORKS LOCALIZATION

Complex network analysis is applied in numerous researches. Features and characteristics of complex networks provide information associated with a network feature called community structure. Naturally, nodes with similar attributes will be more likely to form a community. Community detection is described as the process by which complex network data are analyzed to uncover organizational properties, and structure; and ultimately to enable extraction of useful information. Analysis of Wireless Sensor Networks (WSN) is considered as one of the most important categories of network analysis due to their enormous and emerging applications. Most WSN applications are location-aware, which entails precise localization of the deployed sensor nodes. However, localization of sensor nodes in very dense network is a challenging task. Among various challenges associated with localization of dense WSNs, anchor node selection is shown as a prominent open problem. Optimum anchor selection impacts overall sensor node localization in terms of accuracy and consumed energy. In this thesis, various approaches are developed to address both overlapping and non-overlapping community detection. The proposed approaches target small-size to very large-size networks in near linear time, which is important for very large, densely-connected networks. Performance of the proposed techniques are evaluated over real-world data-sets with up to 106 nodes and syntactic networks via Newman\u27s Modularity and Normalized Mutual Information (NMI). Moreover, the proposed community detection approaches are extended to develop a novel criterion for range-free anchor selection in WSNs. Our approach uses novel objective functions based on nodes\u27 community memberships to reveal a set of anchors among all available permutations of anchors-selection sets. The performance---the mean and variance of the localization error---of the proposed approach is evaluated for a variety of node deployment scenarios and compared with random anchor selection and the full-ranging approach. In order to study the effectiveness of our algorithm, the performance is evaluated over several simulations that randomly generate network configurations. By incorporating our proposed criteria, the accuracy of the position estimate is improved significantly relative to random anchor selection localization methods. Simulation results show that the proposed technique significantly improves both the accuracy and the precision of the location estimation