Multi-scale Community Detection in Temporal Networks Using Spectral Graph Wavelets

Spectral graph wavelets introduce a notion of scale in networks, and are thus used to obtain a local view of the network from each node. By carefully constructing a wavelet filter function for these wavelets, a multi-scale community detection method for monoplex networks has already been developed. This construction takes advantage of the partitioning properties of the network Laplacian. In this paper we elaborate on a novel method which uses spectral graph wavelets to detect multi-scale communities in temporal networks. To do this we extend the definition of spectral graph wavelets to temporal networks by adopting a multilayer framework. We use arguments from Perturbation Theory to investigate the spectral properties of the supra-Laplacian matrix for clustering purposes in temporal networks. Using these properties, we construct a new wavelet filter function, which attenuates the influence of uninformative eigenvalues and centres the filter around eigenvalues which contain information on the coarsest description of prevalent community structures over time. We use the spectral graph wavelets as feature vectors in a connectivity-constrained clustering procedure to detect multi-scale communities at different scales, and refer to this method as Temporal Multi-Scale Community Detection (TMSCD). We validate the performance of TMSCD and a competing methodology on various benchmarks. The advantage of TMSCD is the automated selection of relevant scales at which communities should be sought.