Belief propagation for subgraph detection with imperfect side-information

We propose a local message passing algorithm based on Belief Propagation (BP) to detect a small hidden Erdos-Rényi (ER) subgraph embedded in a larger sparse ER random graph in the presence of side-information. We consider side-information in the form of revealed subgraph nodes called cues, some of which may be erroneous. Namely, the revealed nodes may not all belong to the subgraph, and it is not known to the algorithm a priori which cues are correct and which are incorrect. We show that asymptotically as the graph size tends to infinity, the expected fraction of misclassified nodes approaches zero for any positive value of a parameter λ, which represents the effective Signal-to-Noise Ratio of the detection problem. Previous works on subgraph detection using BP without side-information showed that BP fails to recover the subgraph when λ <; 1/e. Our results thus demonstrate the substantial gains in having even a small amount of side-information.