提出抗干扰的子图检测框架,揭示了真实场景下算法失效的边界。
Robust Detection of Planted Subgraphs in Semi-Random Models
- 设计半随机模型,允许敌手删除子图外边,模拟现实扰动
- 低密度子图在对抗下无法检测,高密度则性能与经典模型一致
- 提出高效鲁棒算法,兼具理论保障与实际可行性
在经典随机图中,对植入子图的检测已建立丰富的统计与计算阈值。然而多数工作假设完全随机生成模型,导致算法在真实世界扰动下可能失效。本文首次研究半随机模型下的植入子图检测问题,其中敌手可在图暴露前移除子图外的边,且统计者不知具体哪些边被删。我们确立该模型下的基本统计极限,发现显著二分现象:对于最大密度为强次对数级的子图,对抗存在时信息论上无法检测——尽管在经典随机模型中部分子图仍可检测;而对超对数密度子图,统计极限几乎不变,最优似然比检验依然稳健。此外,我们设计了一种新的计算高效且鲁棒的检测算法,并提供严格的统计性能保证。本工作建立了首个鲁棒的植入子图检测框架,开启了半随机模型、计算-统计权衡及图推断鲁棒性研究的新方向。
原文摘要 · Abstract (English)
Detection of planted subgraphs in Erdös-Rényi random graphs has been extensively studied, leading to a rich body of results characterizing both statistical and computational thresholds. However, most prior work assumes a purely random generative model, making the resulting algorithms potentially fragile in the face of real-world perturbations. In this work, we initiate the study of semi-random models for the planted subgraph detection problem, wherein an adversary is allowed to remove edges outside the planted subgraph before the graph is revealed to the statistician. Crucially, the statistician remains unaware of which edges have been removed, introducing fundamental challenges to the inference task. We establish fundamental statistical limits for detection under this semi-random model, revealing a sharp dichotomy. Specifically, for planted subgraphs with strongly sub-logarithmic maximum density detection becomes information-theoretically impossible in the presence of an adversary-despite being possible for some planted subgraphs in the classical random model. In stark contrast, for subgraphs with super-logarithmic density, the statistical limits remain essentially unchanged; we prove that the optimal (albeit computationally intractable) likelihood ratio test remains robust. Beyond these statistical boundaries, we design a new computationally efficient and robust detection algorithm, and provide rigorous statistical guarantees for its performance. Our results establish the first robust framework for planted subgraph detection and open new directions in the study of semi-random models, computational-statistical trade-offs, and robustness in graph inference problems.
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