We propose a novel method to find 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. We present the results of the algorithm for computer simulated and real networks and compare them with other approaches. The efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks. 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 description range from the microscopic (degree, clustering coefficient, centrality measures, etc., of individual nodes) to the macroscopic description in terms of statistical properties of the whole network (degree distribution, total clustering coefficient, degree-degree correlations, etc.) [1, 2, 3, 4] . Between these two extremes there is a "mesoscopic" description of networks that tries to explain its community structure. The general notion of community structure in complex networks was first 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, 7, 8, 9 , 10] and its emergence seems to be at the heart of the network formation process [11] . The existing methods intended to devise the community structure in complex networks have been recently reviewed in [10] . All these methods require a definition 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 quantification is still a subject of debate. Some quantitative definitions that came from sociology have been used in recent studies [12] , but in general, the physics community has widely accepted a recent measure for the community structure based on the concept of modularity Q introduced by Newman and Girvan [13] : where e rr are the fraction of links that connect two nodes inside the community r, a r 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 values of Q the most accurate a partition into communities is. The search for the optimal (largest) modularity value is a NP-hard problem due to the fact that the space of possible partitions grows faster than any power of the system size. For this reason, a heuristic search strategy is mandatory to restrict the search space while preserving the optimization goal [14] . Indeed, it is possible to relate the current optimization problem for Q with classical problems in statistical physics, e.g. the spin glass problem of finding the ground state energy [15] , where algorithms inspired in natural optimization processes as simulated annealing [16] and genetic algorithms [17] have been successfully used. In this Letter, we propose a new divisive algorithm that optimizes the modularity Q using an heuristic search based on the Extremal Optimization (EO) algorithm proposed by Boettcher and Percus [18, 19] . This algorithm is inspired in turn in the evolution model of , and basically operates optimizing a global variable by improving extremal local variables that involve coevolutionary avalanches. The performance of EO algorithms have been shown to overcome the efficiency of classical simulated annealing and genetic algorithms providing competitive accuracy [21] . In our case, the global variable to optimize is Q as defined in eq.(1). Thus, the definition of the local variables used in the extremal optimization problem should be related to the contribution of individual nodes i to the summation in eq.(1) given a certain partition into communities where κ r(i) is the number of links that a node i belonging to a community r has with nodes into the same community, and k i is the degree of node i. Note that Q = i q i where i refers to all nodes in the network given a certain partition into communities and L is the total number of links in the network. Eq.(2) provides a
Duch and Arenas REF proposed a divisive algorithm referred as to EO to detect communities in complex networks.
Community detection in complex networks using extremal optimization.
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Mathematics Physics Medicine