Belief Propagation with Side Information for Recovering a Single Community

In this paper, we study the effect of side information on the recovery of a hidden community of size K inside a graph consisting of n nodes with K=o(n). We focus on side information with finite cardinality and bounded (as nrightarrow propto) log-likelihood ratios (LLRs). We calculate tight necessary and sufficient conditions for weak recovery of the labels subject to observation of the graph and side information under belief propagation (BP). Also, we show that BP with side information is strictly inferior to the maximum likelihood detector without side information. Finally, we validate our results through simulations on finite synthetic data-sets that shows the power of our asymptotic results in characterizing the performance even at finite n.