  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 $n$ grows 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. 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 a random guess. The asymptotic MMSE
depends on the covariate signal-to-noise ratio in a more subtle way than the
phase transition threshold does. In the special case of $m=1$, our asymptotic
MMSE formula complements the pioneering work of Deshpande et. al. (2018) which
found the sharp threshold when $m=1$.
