Detection-Recovery Gap for Planted Dense Cycles

Abstract

Planted dense cycles are a type of latent structure that appears in many applications, such as small-world networks in social sciences and sequence assembly in computational biology. We consider a model where a dense cycle with expected bandwidth nτ and edge density p is planted in an Erdős-Rényi graph G(n, q). We characterize the computational thresholds for the associated detection and recovery problems for the class of low-degree polynomial algorithms. In particular, a gap exists between the two thresholds in a certain regime of parameters. For example, if n −3/4 ≪ τ ≪ n −1/2 and p = Cq = Θ(1) for a constant C > 1, the detection problem is computationally easy while the recovery problem is hard for low-degree algorithms.