Stochastic block models with multiple continuous attributes

Abstract

The stochastic block model (SBM) is a probabilistic model for community structure in networks. Typically, only the adjacency matrix is used to perform SBM parameter inference. In this paper, we consider circumstances in which nodes have an associated vector of continuous attributes that are also used to learn the node-to-community assignments and corresponding SBM parameters. While this assumption is not realistic for every application, our model assumes that the attributes associated with the nodes in a network's community can be described by a common multivariate Gaussian model. In this augmented, attributed SBM, the objective is to simultaneously learn the SBM connectivity probabilities with the multivariate Gaussian parameters describing each community. While there are recent examples in the literature that combine connectivity and attribute information to inform community detection, our model is the first augmented stochastic block model to handle multiple continuous attributes. This provides the flexibility in biological data to, for example, augment connectivity information with continuous measurements from multiple experimental modalities. Because the lack of labeled network data often makes community detection results difficult to validate, we highlight the usefulness of our model for two network prediction tasks: link prediction and collaborative filtering. As a result of fitting this attributed stochastic block model, one can predict the attribute vector or connectivity patterns for a new node in the event of the complementary source of information (connectivity or attributes, respectively). We also highlight two biological examples where the attributed stochastic block model provides satisfactory performance in the link prediction and collaborative filtering tasks.

A review of stochastic block models and extensions for graph clustering

“…The model by Gerlach et al (2018) can also be applied to account for the graph between words and documents and the graph between documents simultaneously, in an unifying framework. Finally, the SBM by Stanley et al (2019) that allows continuous attributes on the nodes can potentially be modified to model textual attributes instead.…”

“…Please see Section 7 for further details. Stanley et al (2019) proposed an attribute SBM in which the group memberships Z of the nodes determine both the graph Y and the non-relational attributes…”

“…In the model formulation, in addition to a Bernoulli SBM for an undirected graph, they assumed that the attribute x[p] for node p comes from a mixture of Gaussian distributions with weights θ, which are the probabilities for generating Z p as defined in Section 2.2. By utilising both x and Y together in inferring Z, Stanley et al (2019) found that the two types of information can complement each other, and therefore their attribute SBM is useful for link prediction (Section 6.4).…”

“…25)Daudin et al (2008) also provided an algorithm for the optimisation of the first term on the right hand side. This approximate ICL has been adopted by, for example,Matias et al (2015),Matias and Miele (2017), andStanley et al (2019).…”

Model Description of Similarity-Based Recommendation Systems

“…Then, a similarity matrix defined from the similarity measure is used for the recommendation. Another approach is employing the statistical models, such as stochastic block models [6], that are used to estimate network structures, such as clusters or edge distributions. The learning methods using statistical models often achieve high prediction accuracy in comparison to similarity-based methods.…”