Clustering Network Vertices in Sparse Contextual Multilayer Networks

We consider the problem of learning the latent community structure in a Multi-Layer Contextual Block Model introduced by Ma and Nandy (2021), where the average degree for each of the observed networks is of constant order and establish a sharp detection threshold for the community structure, above which detection is possible asymptotically, while below the threshold no procedure can perform better than random guessing. We further establish that the detection threshold coincides with the threshold for weak recovery of the common community structure using multiple correlated networks and co-variate matrices. Finally, we provide a quasi-polynomial time algorithm to estimate the latent communities in the recovery regime. Our results improve upon the results of Ma and Nandy (2021), which considered the diverging degree regime and recovers the results of Lu and Sen (2020) in the special case of a single network structure.