Detection and Reconstruction of a Random Hypergraph from Noisy Graph Projection

TL;DR

This paper investigates the problem of detecting and reconstructing a random hypergraph from a noisy graph projection, establishing sharp thresholds and revealing a detection-reconstruction gap.

Contribution

It provides the first sharp thresholds for detection and reconstruction in hypergraph projection problems and addresses an open problem from prior research.

Findings

Established sharp thresholds for detection and reconstruction.

Discovered a detection-reconstruction gap phenomenon.

Answered an open problem from previous work.

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 $n^{\delta+o(1)}$, 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 $p=n^{-1+\alpha+o(1)}$ and each edge not in the projection is added with probability $q=n^{-1+\beta+o(1)}$. For all constant $d$, we establish sharp thresholds for both detection (distinguishing the noisy projection from an Erd\H{o}s-R\'enyi 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+}.