Detecting Arbitrary Planted Subgraphs in Random Graphs
Dor Elimelech, Wasim Huleihel
TL;DR
This work develops a unified theory for detecting an arbitrary planted subgraph Γ_n in Erdős–Rényi graphs by analyzing both information-theoretic limits and computational feasibility. It introduces the union-planting model, derives dense-regime thresholds governed by χ^2(p||q) and graph-structure statistics μ(Γ_n), |e(Γ_n)|, and d_max(Γ_n), and demonstrates when a computational-statistical gap can arise via low-degree polynomial arguments. The results extend to sparse and critical regimes, providing general bounds and identifying sharp phase transitions in several regimes, including sub- and super-log-density cases. The study also presents practical, polynomial-time detectors (count, degree, scan tests) with conditions under which they succeed, and it develops a comprehensive lower-bound framework built on polynomial decompositions and vertex-cover analyses to characterize when detection is statistically impossible, thereby framing the landscape of planted-subgraph detection in a broad and unified way.
Abstract
The problems of detecting and recovering planted structures/subgraphs in Erdős-Rényi 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 $Γ= Γ_n$ in an Erdős-Rényi random graph $\mathcal{G}(n, q_n)$, where the edge probability within $Γ$ 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 $Γ$, and provide conditions under which a computational-statistical gap arises. Most notably, these thresholds depend on $Γ$ 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 $q_n = Θ(n^{-α})$ and $p_n-q_n =Θ(q_n)$, with $α\in[0,2]$, as well as the critical regime where $p_n=1-o(1)$ and $q_n = Θ(n^{-α})$, both of which have been widely studied, for specific choices of $Γ$. For these regimes, we show that our bounds are tight for all planted subgraphs investigated in the literature thus far\textemdash{}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 $q_n$.