Information-Theoretic Thresholds for Planted Dense Cycles

TL;DR

This paper investigates the fundamental limits of detecting and recovering a hidden dense cycle in a random graph model, revealing gaps between what is information-theoretically possible and what current algorithms can achieve.

Contribution

It characterizes the exact information-theoretic thresholds for detection and recovery of planted dense cycles, highlighting the existence of statistical-to-computational gaps.

Findings

Thresholds depend on graph size, cycle bandwidth, and signal-to-noise ratio.

Information-theoretic limits differ from computational thresholds, indicating gaps.

Results justify the presence of statistical-to-computational gaps in this problem.

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 $n \tau$, 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.