  Community detection refers to the problem of clustering the nodes of a
network into groups. Existing inferential methods for community structure
mainly focus on unweighted (binary) networks. Many real-world networks are
nonetheless weighted and a common practice is to dichotomize a weighted network
to an unweighted one which is known to result in information loss. Literature
on hypothesis testing in the latter situation is still missing. In this paper,
we study the problem of testing the existence of community structure in
weighted networks. Our contributions are threefold: (a). We use the (possibly
infinite-dimensional) exponential family to model the weights and derive the
sharp information-theoretic limit for the existence of consistent test. Within
the limit, any test is inconsistent; and beyond the limit, we propose a useful
consistent test. (b). Based on the information-theoretic limits, we provide the
first formal way to quantify the loss of information incurred by dichotomizing
weighted graphs into unweighted graphs in the context of hypothesis testing.
(c). We propose several new and practically useful test statistics. Simulation
study show that the proposed tests have good performance. Finally, we apply the
proposed tests to an animal social network.
