# Information-Theoretic Thresholds for Planted Dense Cycles

CoRR（2024）

Abstract

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 τ, 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, τ, and an edge-wise
signal-to-noise ratio λ. 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.

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