Nonparametric weighted stochastic block models

Many network systems lack a natural low-dimensional embedding from which we can readily extract their most prominent large-scale features. Instead, we have to infer this information from data, typically by decomposing the observed network into modules [1]. A principled approach to perform this task is to formulate generative models that allow this modular decomposition to be found via statistical inference [2]. The most fundamental model used for this purpose is the stochastic block model (SBM) [3], which groups nodes according to their probabilities of connection to the rest of the network. However, a central limitation of most SBM implementations is that they are defined strictly for simple or multigraphs. This means that they do not incorporate extra information on the edges, which are typically present in a variety of systems, and are required for an accurate representation of their structure. For example, to the existence of a route between two airports is associated a distance, to the biomass flow between two species in a food web is associated a flow magnitude, etc. In this work, we develop variations of the SBM that allow for this type of information on the edges to be incorporated into the network model and guide the partition of the nodes into groups in a statistically meaningful way. We follow the same basic idea put forth by Aicher et al. [4], who adapted the SBM to weighted networks by including edge values as additional covariates.