Detecting Arbitrary Planted Subgraphs in Random Graphs
This provides a general theory for planted subgraph detection, addressing a long-standing gap in the literature that previously focused on specific cases, with implications for statistical inference and algorithm design in random graph models.
The paper tackles the problem of detecting arbitrary planted subgraphs in Erdős-Rényi random graphs, establishing tight information-theoretic and computational thresholds that depend on key graph properties like number of edges, maximum degree, and maximum subgraph density, and showing these bounds are tight across dense, sparse, and critical regimes for many subgraphs.
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 $q_n$.