One of the most fundamental problems in network study is community detection.
The stochastic block model (SBM) is one widely used model for network data with
different estimation methods developed with their community detection
consistency results unveiled. However, the SBM is restricted by the strong
assumption that all nodes in the same community are stochastically equivalent,
which may not be suitable for practical applications. We introduce a pairwise
covariates-adjusted stochastic block model (PCABM), a generalization of SBM
that incorporates pairwise covariate information. We study the maximum
likelihood estimates of the coefficients for the covariates as well as the
community assignments. It is shown that both the coefficient estimates of the
covariates and the community assignments are consistent under suitable sparsity
conditions. Spectral clustering with adjustment (SCWA) is introduced to
efficiently solve PCABM. Under certain conditions, we derive the error bound of
community estimation under SCWA and show that it is community detection
consistent. PCABM compares favorably with the SBM or degree-corrected
stochastic block model (DCBM) under a wide range of simulated and real networks
when covariate information is accessible.Comment: 41 pages, 6 figure