Robust Detection of Planted Subgraphs in Semi-Random Models

2025-08-04

AI Summary

This work studies robust detection of planted subgraphs under a semi-random model, where an adversary may arbitrarily delete edges outside the planted subgraph—rendering standard detection methods ineffective. We first formalize this semi-random detection model and identify a sharp statistical detectability phase transition at the maximum density threshold. Using information-theoretic analysis, likelihood ratio testing, and adversarial modeling, we characterize the fundamental statistical limits of detection and prove the robustness of classical detection boundaries in the high-density regime. Furthermore, we design the first efficient robust detection algorithm with rigorous theoretical guarantees. Our work establishes the first systematic framework for robust detection in graph-structured inference, providing a novel paradigm and theoretical foundation for the computational-statistical tradeoff in adversarial settings.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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 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.

Problem

• Study detection of planted subgraphs in semi-random graphs
• Analyze robustness against adversarial edge removal
• Establish statistical limits and efficient detection algorithms

Innovation

• Study semi-random models for subgraph detection
• Establish statistical limits under adversarial conditions
• Develop robust and efficient detection algorithm