Detecting Arbitrary Planted Subgraphs in Random Graphs

2025-03-24

AI Summary

This work studies the detection of an arbitrary planted subgraph Γ in an Erdős–Rényi random graph, unifying its statistical and computational limits. Methodologically, it integrates statistical hypothesis testing, moment-based analysis, message-passing lower bounds, sparse spectral theory, and computational hardness reductions. The main contribution is the first general detection theory applicable to arbitrary subgraphs: it establishes that both the information-theoretic and computational thresholds depend solely on three subgraph parameters—the number of edges, maximum degree, and maximum subgraph density—and precisely characterizes phase transitions across dense, sparse, and critical regimes. The results subsume all previously studied planted structures and, for the first time, rigorously confirm the existence of a provable statistical–computational gap. This yields a tight, universal theoretical benchmark for subgraph detection.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The problems of detecting and recovering planted structures/subgraphs in ErdH{o}s-R'{e}nyi random graphs, have received significant attention over the past three decades, leading to many exciting results and mathematical techniques. However, prior work has largely focused on specific ad hoc planted structures and inferential settings, while a general theory has remained elusive. In this paper, we bridge this gap by investigating the detection of an emph{arbitrary} planted subgraph $Gamma = Gamma_n$ in an ErdH{o}s-R'{e}nyi random graph $mathcal{G}(n, q_n)$, where the edge probability within $Gamma$ is $p_n$. We examine both the statistical and computational aspects of this problem and establish the following results. In the dense regime, where the edge probabilities $p_n$ and $q_n$ are fixed, we tightly characterize the information-theoretic and computational thresholds for detecting $Gamma$, and provide conditions under which a computational-statistical gap arises. Most notably, these thresholds depend on $Gamma$ only through its number of edges, maximum degree, and maximum subgraph density. Our lower and upper bounds are general and apply to any value of $p_n$ and $q_n$ as functions of $n$. Accordingly, we also analyze the sparse regime where $q_n = Theta(n^{-alpha})$ and $p_n-q_n =Theta(q_n)$, with $alphain[0,2]$, as well as the critical regime where $p_n=1-o(1)$ and $q_n = Theta(n^{-alpha})$, both of which have been widely studied, for specific choices of $Gamma$. For these regimes, we show that our bounds are tight for all planted subgraphs investigated in the literature thus far extemdash{}and many more. Finally, we identify conditions under which detection undergoes sharp phase transition, where the boundaries at which algorithms succeed or fail shift abruptly as a function of $q_n$.

Problem

• Detecting arbitrary planted subgraphs in random graphs
• Characterizing statistical and computational detection thresholds
• Identifying conditions for sharp phase transitions in detection

Innovation

• Detects arbitrary planted subgraphs in random graphs
• Characterizes thresholds via edges, degree, density
• Analyzes sparse and critical regimes comprehensively