Community detection in complex networks using Extremal Optimization

We propose a novel method to flnd the community structure in complex networks based on an extremal optimization of the value of modularity. The method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure. We present the results of the algorithm for computer simulated and real networks and compare them with other approaches. The e–ciency and accuracy of the method make it feasible to be used for the accurate identiflcation of community structure in large complex networks. PACS

numbers: The description of the structure of complex networks has been one of the focus of attention of the physicist’s community in the recent years. The levels of descrip- tion range from the microscopic (degree, clustering coef- flcient, centrality measures, etc., of individual nodes) to the macroscopic description in terms of statistical proper- ties of the whole network (degree distribution, total clus- tering coe–cient, degree-degree correlations, etc.) [1{4]. Between these two extremes there is a "mesoscopic" de- scription of networks that tries to explain its community structure. The general notion of community structure in complex networks was flrst pointed out in the physics literature by Girvan and Newman [5], and refers to the fact that nodes in many real networks appear to group in subgraphs in which the density of internal connections is larger than the connections with the rest of nodes in the network. The community structure has been empirically found in many real technological, biological and social networks [6{12] and its emergence seems to be at the heart of the network formation process [13]. The existing methods intended to devise the commu- nity structure in complex networks have been recently reviewed in [10]. All these methods require a deflnition of community that imposes the limit up to which a group should be considered a community. However, the concept of community itself is qualitative: nodes must be more connected within its community than with the rest of the network, and its quantiflcation is still a subject of debate. Some quantitative deflnitions that came from sociology have been used in recent studies [14], but in general, the physics community has widely accepted a measure for the community structure based on the concept of modularity Q introduced by Newman and Girvan [15]: Q = X r (err ¡a2r) (1) where err are the fraction of links that connect two nodes inside the community r, ar the fraction of links that have one or both vertices inside of the community r, and the sum extends to all communities r in a given network. Note that this measure provides a way to determine if a certain mesoscopic description of the graph in terms of communities is more or less accurate. The larger the val- ues of Q the most accurate a partition into communities is. The search for the optimal (largest) modularity value seems to be a NP-hard problem due to the fact...

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