Detecting a planted community in an inhomogeneous random graph

We study the problem of detecting whether an inhomogeneous random graph contains a planted community. Specifically, we observe single realization graph. Under null hypothesis, this is sample from graph, whereas under alternative, there exists small subgraph where edge probabilities are increased by multiplicative scaling factor. present scan test that able to detect presence such community, even when community very and underlying inhomogeneous. also derive information theoretic lower bound for which shows in some regimes almost asymptotically optimal. illustrate our results through examples numerical experiments.