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We present a method based on the orthogonal symmetric non-negative matrixtri-factorization of the normalized Laplacian matrix for community detection incomplex networks. While the exact factorization of a given order may not existand is NP hard to compute, we obtain an approximate factorization by solving anoptimization problem. We establish the connection of the factors obtainedthrough the factorization to a non-negative basis of an invariant subspace ofthe estimated matrix, drawing parallel with the spectral clustering. Using suchfactorization for clustering in networks is motivated by analyzing ablock-diagonal Laplacian matrix with the blocks representing the connectedcomponents of a graph. The method is shown to be consistent for communitydetection in graphs generated from the stochastic block model and the degreecorrected stochastic block model. Simulation results and real data analysisshow the effectiveness of these methods under a wide variety of situations,including sparse and highly heterogeneous graphs where the usual spectralclustering is known to fail. Our method also performs better than the state ofthe art in popular benchmark network datasets, e.g., the political web blogsand the karate club data.

Orthogonal symmetric non-negative matrix factorization under the stochastic block model