Community Detection With Contextual Multilayer Networks

In this paper, we study community detection when observe $m$ sparse networks and a high dimensional covariate matrix, all encoding the same structure among notation="LaTeX">$n$ subjects. asymptotic regime where number of features notation="LaTeX">$p$ subjects grow proportionally, derive an exact formula minimum mean square error (MMSE) for estimating common in balanced two block case using orchestrated approximate message passing algorithm. The implies necessity integrating information from multiple data sources. Consequently, it induces sharp threshold phase transition between (i.e., weak recovery) is possible no procedure performs better than random guess. MMSE depends on signal-to-noise ratio more subtle way threshold. special notation="LaTeX">$m=1$ , our complements pioneering work Deshpande et al., (2018) which found . A practical variant theoretically justified algorithm with spectral initialization leads to estimator whose empirical MSEs closely theoretical predictions over simulated examples.