  We consider the problem of community detection in overlapping weighted
networks, where nodes can belong to multiple communities and edge weights can
be finite real numbers. To model such complex networks, we propose a general
framework - the mixed membership distribution-free (MMDF) model. MMDF has no
distribution constraints of edge weights and can be viewed as generalizations
of some previous models, including the well-known mixed membership stochastic
blockmodels. Especially, overlapping signed networks with latent community
structures can also be generated from our model. We use an efficient spectral
algorithm with a theoretical guarantee of convergence rate to estimate
community memberships under the model. We also propose the fuzzy weighted
modularity to evaluate the quality of community detection for overlapping
weighted networks with positive and negative edge weights. We then provide a
method to determine the number of communities for weighted networks by taking
advantage of our fuzzy weighted modularity. Numerical simulations and real data
applications are carried out to demonstrate the usefulness of our mixed
membership distribution-free model and our fuzzy weighted modularity.
