Recovery of Planted Subgraphs

2026-07-01

AI Summary

This work investigates the fundamental limits of exactly recovering an arbitrarily planted subgraph in dense Erdős–Rényi random graphs. Addressing the challenge posed by general subgraph structures that existing methods struggle to handle, the study introduces a novel graph-theoretic measure—the minimum maximal subgraph density—which precisely characterizes the sharp statistical threshold for exact recovery and reveals a statistically feasible yet computationally hard phase transition regime. A polynomial-time spectral algorithm is developed for recovery, and matching computational lower bounds are established via the low-degree polynomial framework. The results apply universally to arbitrary planted subgraphs and further extend to semi-random models and weak recovery settings, thereby providing a comprehensive delineation of the boundary between statistical and computational complexity.

Abstract

Understanding the fundamental limits of recovering planted subgraphs in random graphs is a central challenge in high-dimensional statistics and theoretical computer science. While existing work has largely focused on special subgraph families such as cliques, bicliques, or dense blocks, the exact recovery of a general planted subgraph in Erdős--Rényi random graphs remains poorly understood. In this paper, we study the exact recovery of an arbitrary planted subgraph $Γ= Γ_n$ embedded in a dense Erdős--Rényi random graph $\mathcal{G}(n,q_n)$, where edges within $Γ$ are present independently with probability $p_n > q_n$. Our main results identify sharp conditions under which exact recovery is possible with high probability, and we establish matching lower bounds showing the necessity of these conditions. The resulting statistical threshold is characterized by a new graph-theoretic quantity, which we term the \emph{minimal maximum subgraph density}. This quantity is defined as the maximum subgraph density of the smallest induced balanced subgraph of $Γ$. We then turn to the problem of recovery under polynomial-time constraints. We propose a computationally efficient recovery algorithm that applies to arbitrary planted subgraphs and analyze its performance in terms of certain spectral properties of the adjacency matrix. In addition, we derive computational lower bounds for recovery using the low-degree polynomial framework, establishing regimes where recovery is statistically possible but computationally hard. Finally, we consider several extensions of our setting, including recovery in semi-random models and weaker notions of recovery.

Problem

• planted subgraph
• exact recovery
• random graphs
• statistical limits
• Erdős–Rényi

Innovation

• planted subgraph recovery
• minimal maximum subgraph density
• statistical-computational gap
• spectral algorithm
• low-degree polynomial method