A Dirichlet stochastic block model for composition-weighted networks

Network data are prevalent in applications where individual entities interact with each other, and often these interactions have associated weights representing the strength of association. Clustering such weighted network data is a common task, which involves identifying groups of nodes that display similarities in the way they interact. However, traditional clustering methods typically use edge weights in their raw form, overlooking that the observed weights are influenced by the nodes' capacities to distribute weights along the edges. This can lead to clustering results that primarily reflect nodes' total weight capacities rather than the specific interactions between them. One way to address this issue is to analyse the strengths of connections in relative rather than absolute terms, by transforming the relational weights into a compositional format. This approach expresses each edge weight as a proportion of the sending or receiving weight capacity of the respective node. To cluster these data, a Dirichlet stochastic block model tailored for composition-weighted networks is proposed. The model relies on direct modelling of compositional weight vectors using a Dirichlet mixture, where parameters are determined by the cluster labels of sender and receiver nodes. Inference is implemented via an extension of the classification expectation-maximisation algorithm, expressing the complete data likelihood of each node as a function of fixed cluster labels of the remaining nodes. A model selection criterion is derived to determine the optimal number of clusters. The proposed approach is validated through simulation studies, and its practical utility is illustrated on two real-world networks.