  The mixed membership stochastic blockmodel (MMSB) is a popular framework for
community detection and network generation. It learns a low-rank mixed
membership representation for each node across communities by exploiting the
underlying graph structure. MMSB assumes that the membership distributions of
the nodes are independently drawn from a Dirichlet distribution, which limits
its capability to model highly correlated graph structures that exist in
real-world networks. In this paper, we present a flexible richly structured
MMSB model, \textit{Struct-MMSB}, that uses a recently developed statistical
relational learning model, hinge-loss Markov random fields (HL-MRFs), as a
structured prior to model complex dependencies among node attributes,
multi-relational links, and their relationship with mixed-membership
distributions. Our model is specified using a probabilistic programming
templating language that uses weighted first-order logic rules, which enhances
the model's interpretability. Further, our model is capable of learning latent
characteristics in real-world networks via meaningful latent variables encoded
as a complex combination of observed features and membership distributions. We
present an expectation-maximization based inference algorithm that learns
latent variables and parameters iteratively, a scalable stochastic variation of
the inference algorithm, and a method to learn the weights of HL-MRF structured
priors. We evaluate our model on six datasets across three different types of
networks and corresponding modeling scenarios and demonstrate that our models
are able to achieve an improvement of 15\% on average in test log-likelihood
and faster convergence when compared to state-of-the-art network models.
