Community Detection With Side Information: Exact Recovery Under the Stochastic Block Model

The community detection problem involves making inferences about node labels in a graph, based on observing the graph edges. This paper studies the effect of additional, nongraphical side information on the phase transition of exact recovery in the binary stochastic block model with
$n$
nodes. When side information consists of noisy labels with error probability
$\alpha$
, it is shown that phase transition is improved if and only if
$\log (\frac{1-\alpha }{\alpha })=\Omega (\log (n))$
. When side information consists of revealing a fraction
$1-\epsilon$
of the labels, it is shown that phase transition is improved if and only if
$\log (1/\epsilon)=\Omega (\log (n))$
. For a more general side information consisting of
$K$
features, two scenarios are studied, first, 
$K$
is fixed while the likelihood of each feature with respect to corresponding node label evolves with
$n$
, and, second, the number of features
$K$
varies with
$n$
but the likelihood of each feature is fixed. In each case, we find when side information improves the exact recovery phase transition and by how much. In the process of deriving inner bounds, a variation of an efficient algorithm is proposed for community detection with side information that uses a partial recovery algorithm combined with a local improvement procedure.