Robust Detection of Planted Subgraphs in Semi-Random Models

TL;DR

This paper introduces a semi-random model for planted subgraph detection, revealing fundamental limits and proposing a robust detection algorithm, thus advancing understanding of robustness and computational-statistical trade-offs in graph inference.

Contribution

It establishes the first robust framework for planted subgraph detection under semi-random models, analyzing statistical limits and designing a computationally efficient robust algorithm.

Findings

Detection becomes impossible for subgraphs with strongly sub-logarithmic density under adversarial edge removal.

For subgraphs with super-logarithmic density, the likelihood ratio test remains effective and robust.

A new efficient algorithm is proposed with proven statistical guarantees in the semi-random setting.

Abstract

Detection of planted subgraphs in Erd\"os-R\'enyi 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.