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Community detection in graphs has been the subject of many algorithms. Recentmethods want to optimize a modularity function which shows a maximum ofrelationships within communities and found a minimum of inter-communityrelations. these algorithms are applied to unipartite, multipartite anddirected graphs. However, given the NP-completeness of the problem, thesealgorithms are heuristics that do not guarantee an optimum. In this paper weintroduce an algorithm which, based on an approximate solution obtained througha efficient detection algorithm, modifie it to achieve a local optimum based ona function. this reassignment function is a potential function and thereforethe computed optimum is a Nash equilibrium. We supplement our method with anoverlap function that allows to have simultaneously the two detection modes.Several experiments show the interest of our approach.

Optimisation dans la d\'etection de communaut\'es recouvrantes et \'equilibre de Nash