  The study of network data in the social and health sciences frequently
concentrates on two distinct tasks (1) detecting community structures among
nodes and (2) associating covariate information to edge formation. In much of
this data, it is likely that the effects of covariates on edge formation differ
between communities (e.g. age might play a different role in friendship
formation in communities across a city). In this work, we introduce a latent
space network model where coefficients associated with certain covariates can
depend on latent community membership of the nodes. We show that ignoring such
structure can lead to either over- or under-estimation of covariate importance
to edge formation and propose a Markov Chain Monte Carlo approach for
simultaneously learning the latent community structure and the community
specific coefficients. We leverage efficient spectral methods to improve the
computational tractability of our approach.
