Detection and Reconstruction of a Random Hypergraph from Noisy Graph Projection

TL;DR

The paper investigates recovering a planted $d$-uniform hypergraph from its noisy graph projection, formalizing a model with parameters $s=n^{-(d-1)+δ+o(1)}$, $p=n^{-1+α+o(1)}$, and $q=n^{-1+β+o(1)}$. It derives sharp detection and reconstruction thresholds, unveiling a detection-reconstruction gap where detection is easier than reconstruction in broad regimes. The authors introduce and analyze an edge-counting detector for detection, a clique-based estimator for partial/almost-exact reconstruction, and information-theoretic tools (Fano-type bounds, Pinsker/Talagrand arguments) to establish impossibility results. They also discuss a modified null distribution that partially mitigates the gap but cannot fully remove it. Overall, the work advances understanding of when and how the original higher-order structure can be inferred from pairwise observations under two-way noise, answering open questions in the noisy-projection setting and delineating the regimes of feasible recovery.

Abstract

For a $d$-uniform random hypergraph on $n$ vertices in which hyperedges are included i.i.d.\ so that the average degree in the hypergraph is $n^{δ+o(1)}$, the projection of such a hypergraph is a graph on the same $n$ vertices where an edge connects two vertices if and only if they belong to a same hyperedge. In this work, we study the inference problem where the observation is a \emph{noisy} version of the graph projection where each edge in the projection is kept with probability $p=n^{-1+α+o(1)}$ and each edge not in the projection is added with probability $q=n^{-1+β+o(1)}$. For all constant $d$, we establish sharp thresholds for both detection (distinguishing the noisy projection from an Erdős-Rényi random graph with edge density $q$) and reconstruction (estimating the original hypergraph). Notably, our results reveal a \emph{detection-reconstruction gap} phenomenon in this problem. Our work also answers a problem raised in \cite{BGPY25+}.

Figures (1)

• Figure 1: Summary of the detection-reconstruction gap when $p=\Theta(1)$. Red regime $I$: reconstruction of $\mathcal{H}$ is possible; Blue regime $II$: detection with a better null distribution $\widetilde{\mathbb Q}$ (that matches the average edge-density) is possible, but reconstruction of $\mathcal{H}$ is impossible; Yellow regime $III$: detection with the simple null distribution $\mathbb Q$ is possible, but detection with a better null distribution $\widetilde{\mathbb Q}$ is unknown.

Theorems & Definitions (22)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.5
• Definition 2.6
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9
• Lemma 4.1