Using Stochastic Block Models for Community Detection: The issue of edge-connectivity

A relevant, sometimes overlooked, quality criterion for communities in graphs is that they should be well-connected in addition to being edge-dense. Prior work has shown that leading community detection methods can produce poorly-connected communities, and some even produce internally disconnected communities. A recent study by Park et al. in Complex Networks and their Applications 2024 showed that this problem is evident in clusterings from three Stochastic Block Models (SBMs) in graph-tool, a popular software package. To address this issue, Park et al. presented a simple technique, Well-Connected Clusters (WCC), that repeatedly finds and removes small edge cuts of size at most $\log_{10}n$ in clusters, where $n$ is the number of nodes in the cluster, and showed that treatment of graph-tool SBM clusterings with WCC improves accuracy. Here we examine the question of cluster connectivity for clusterings computed using other SBM software or nested SBMs within graph-tool. Our study, using a wide range of real-world and synthetic networks, shows that all tested SBM clustering methods produce communities that are disconnected, and that graph-tool improves on PySBM. We provide insight into why graph-tool degree-corrected SBM clustering produces disconnected clusters by examining the description length formula it uses, and explore the impact of modifications to the description length formula. Finally, we show that WCC provides an improvement in accuracy for both flat and nested SBMs and establish that it scales to networks with millions of nodes.