  We study a random graph model for small-world networks which are ubiquitous
in social and biological sciences. In this model, a dense cycle of expected
bandwidth $n \tau$, representing the hidden one-dimensional geometry of
vertices, is planted in an ambient random graph on $n$ vertices. For both
detection and recovery of the planted dense cycle, we characterize the
information-theoretic thresholds in terms of $n$, $\tau$, and an edge-wise
signal-to-noise ratio $\lambda$. In particular, the information-theoretic
thresholds differ from the computational thresholds established in a recent
work for low-degree polynomial algorithms, thereby justifying the existence of
statistical-to-computational gaps for this problem.
