Community mining with graph filters for correlation matrices

Communities are an important type of structure in networks. Graph filters, such as wavelet filterbanks, have been used to detect such communities as groups of nodes more densely connected together than with the outsiders. When dealing with times series, it is possible to build a relational network based on the correlation matrix. However, in such a network, weights assigned to each edge have different properties than those of usual adjacency matrices. As a result, classical community detection methods based on modularity optimization are not consistent and the modularity needs to be redefined to take into account the structure of the correlation from random matrix theory. Here, we address how to detect communities from correlation matrices, by filtering global modes and random parts using properties that are specific to the distribution of correlation eigenvalues. Based on a Louvain approach, an algorithm to detect multiscale communities is also developed, which yields a weighted hierarchy of communities. The implementation of the method using graph filters is also discussed.