Information-Theoretic Thresholds for Planted Dense Cycles

TL;DR

This work establishes information-theoretic thresholds for detecting and recovering a planted dense cycle in a random graph, characterized by the parameters $n$, $tau$, and $lambda = (p-q)^2/(r(1-r))$. The authors show that when $n tau lambda \to 0$ no weak detection or weak recovery is possible, while under $n tau lambda / \log n \to \infty$ and $n tau (p-r)/\log n \to \infty$ strong detection and recovery are achievable, with the critical threshold at roughly $n tau lambda = \tilde{\Theta}(1)$. To prove the lower bounds, they develop a conditional second moment method and a delicate concentration analysis for $U$-statistics, and for recovery they connect MMSE to mutual information via an interpolation framework $\mathcal{P}_\theta$. Upper bounds are obtained via a geometric optimization over realizable cycles, yielding matching guarantees under the same scaling, up to logarithmic factors. The results demonstrate a statistical-to-computational gap relative to low-degree polynomial algorithms, highlighting instances where information-theoretic feasibility does not coincide with efficient computability in planted dense-cycle models.

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.

Figures (1)

• Figure 1: The statistical-to-computational gaps for detection and recovery of a planted dense cycle with $p = n^{-a}$, $\tau = n^{-b}$, and $C q \leq p \leq C' q$. Both tasks become more difficult as $a$ and $b$ increase. Black solid line: information-theoretic threshold for both detection and recovery. Red dotted line: computational (low-degree) threshold for detection. Blue dashed line: computational threshold for recovery.

Theorems & Definitions (48)

• Definition 1.1: Model $\mathcal{P}$, planted dense cycle
• Definition 1.2: Model $\mathcal{Q}$, Erdős--Rényi graph
• Definition 1.3: Detection
• Definition 1.4: Recovery
• Theorem 1.5: Information-theoretic thresholds
• proof : Proof of Theorem \ref{['thm:main result']}
• Remark 1.6: Informal summary of computational thresholds maoDetectionRecoveryGapPlanted2023
• Theorem 2.1
• Lemma 2.2
• proof