Community Detection With Contextual Multilayer Networks

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