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We show here that the problem of maximizing a family of quantitativefunctions, encompassing both the modularity (Q-measure) and modularity density(D-measure), for community detection can be uniformly understood as acombinatoric optimization involving the trace of a matrix called modularityLaplacian. Instead of using traditional spectral relaxation, we applyadditional nonnegative constraint into this graph clustering problem and designefficient algorithms to optimize the new objective. With the explicitnonnegative constraint, our solutions are very close to the ideal communityindicator matrix and can directly assign nodes into communities. Thenear-orthogonal columns of the solution can be reformulated as the posteriorprobability of corresponding node belonging to each community. Therefore, theproposed method can be exploited to identify the fuzzy or overlappingcommunities and thus facilitates the understanding of the intrinsic structureof networks. Experimental results show that our new algorithm consistently,sometimes significantly, outperforms the traditional spectral relaxationapproaches.

Modularity functions maximization with nonnegative relaxation facilitates community detection in networks