  We show here that the problem of maximizing a family of quantitative
functions, encompassing both the modularity (Q-measure) and modularity density
(D-measure), for community detection can be uniformly understood as a
combinatoric optimization involving the trace of a matrix called modularity
Laplacian. Instead of using traditional spectral relaxation, we apply
additional nonnegative constraint into this graph clustering problem and design
efficient algorithms to optimize the new objective. With the explicit
nonnegative constraint, our solutions are very close to the ideal community
indicator matrix and can directly assign nodes into communities. The
near-orthogonal columns of the solution can be reformulated as the posterior
probability of corresponding node belonging to each community. Therefore, the
proposed method can be exploited to identify the fuzzy or overlapping
communities and thus facilitates the understanding of the intrinsic structure
of networks. Experimental results show that our new algorithm consistently,
sometimes significantly, outperforms the traditional spectral relaxation
approaches.
