  We investigate how to select the number of communities for weighted networks
without a full likelihood modeling. First, we propose a novel weighted
degree-corrected stochastic block model (DCSBM), in which the mean adjacency
matrix is modeled as the same as in standard DCSBM, while the variance profile
matrix is assumed to be related to the mean adjacency matrix through a given
variance function. Our method of selection the number of communities is based
on a sequential testing framework, in each step the weighed DCSBM is fitted via
some spectral clustering method. A key step is to carry out matrix scaling on
the estimated variance profile matrix. The resulting scaling factors can be
used to normalize the adjacency matrix, from which the testing statistic is
obtained. Under mild conditions on the weighted DCSBM, our proposed procedure
is shown to be consistent in estimating the true number of communities.
Numerical experiments on both simulated and real network data also demonstrate
the desirable empirical properties of our method.
