Information-Theoretic Thresholds for Planted Dense Cycles

We study a random graph model for small-world networks which are ubiquitousin social and biological sciences. In this model, a dense cycle of expectedbandwidth n τ, representing the hidden one-dimensional geometry ofvertices, is planted in an ambient random graph on n vertices. For bothdetection and recovery of the planted dense cycle, we characterize theinformation-theoretic thresholds in terms of n, τ, and an edge-wisesignal-to-noise ratio λ. In particular, the information-theoreticthresholds differ from the computational thresholds established in a recentwork for low-degree polynomial algorithms, thereby justifying the existence ofstatistical-to-computational gaps for this problem.