Geometric Multiscale Community Detection: Markov Stability and Vector Partitioning

Multiscale community detection can be viewed from a dynamical perspective within the Markov Stability framework, which uses the diffusion of a Markov process on the graph to uncover intrinsic network substructures across all scales. Here we reformulate multiscale community detection as a max-sum length vector partitioning problem with respect to the set of time-dependent node vectors expressed in terms of eigenvectors of the transition matrix. This formulation provides a geometric interpretation of Markov Stability in terms of a time-dependent spectral embedding, where the Markov time acts as an inhomogeneous geometric resolution factor that zooms the components of the node vectors at different rates. Our geometric formulation encompasses both modularity and the multi-resolution Potts model, which are shown to correspond to vector partitioning in a pseudo-Euclidean space, and is also linked to spectral partitioning methods, where the number of eigenvectors used corresponds to the dimensionality of the underlying embedding vector space. Inspired by the Louvain optimisation for community detection, we then propose an algorithm based on a graph-theoretical heuristic for the vector partitioning problem. We apply the algorithm to the spectral optimisation of modularity and Markov Stability community detection. The spectral embedding based on the transition matrix eigenvectors leads to improved partitions with higher information content and higher modularity than the eigen-decomposition of the modularity matrix. We illustrate the results with random network benchmarks.