Information-Theoretic Thresholds for Planted Dense Cycles

• URL: http://arxiv.org/abs/2402.00305v1
• Date: Thu, 1 Feb 2024 03:39:01 GMT
• Title: Information-Theoretic Thresholds for Planted Dense Cycles
• Authors: Cheng Mao, Alexander S. Wein, Shenduo Zhang
• Abstract summary: We study a random graph model for small-world networks which are ubiquitous in social and biological sciences. 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 $lambda$.
• Score: 52.076657911275525
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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