Equipping SBMs eith RBMs: A Statistical model of Networks with Covariates

Gururaj K

Problem statement and Overview

Network indexed data often contains not only connectivity information but also observable node and/or edge covariates. Capturing the unknown relation between node covariates and the presence of edges is therefore very useful and is the primary motive of this work. We propose a simple but general modeling framework that combines the strengths of Stochastic Block Models and Restricted Boltzmann Machines. This captures the relation between node covariates and community structure, without making an explicit assumption on the causal direction between these two quantities, achieving our aim.

Contribution and Results

• We propose two simple and flexible generative models for modeling Attributed Networks with non-overlapping and mixed membership communities (RB-SBM and RB-MMSBM resp) by combining variants of RBMs with SBMs. • We derive efficient inference methods for the proposed models. Each iteration of the Inference algorithm for RB-SBM runs in linear time with respect to the number of nodes and edges thus making it scalable. • We empirically validate the proposed models on the task of community detection & link prediction and demonstrate through series of quantitative experiments and qualitative case studies that they can provide interpretable insights about the data. • Our approach (RB-SBM model) outperforms existing approaches on Cora and Citeseer networks in terms of NMI score with respect to known ground truth community memberships. RB-MMSBM model recovers more meaningful communities compared to baseline MMSBM model on Lazega Lawyers dataset. • RB-SBM model achieves comparable results in link prediction task on Cora and Citeseer networks when compared with deep learning approaches such as GAE, VGAE despite having significantly fewer parameters. • We study the effect of Non-informative covariates and suggest modifications to mitigate short-comings. • We also analyze the Degree Corrected variant of RB-SBM model and compare the results.

Conclusion and Future Work

We believe that our proposed model serves as a stepping-stone for generalization of SBMs for networks with node and link covariates. Models similar in spirit to RBM can be developed to incorporate link covariates as well leading to better modeling of Networks with Covariates.