Thresholds for Reconstruction of Random Hypergraphs From Graph Projections

TL;DR

This paper investigates the conditions under which a random hypergraph can be reconstructed from its graph projection, providing precise thresholds for certain cases and bounds for others, with implications for hypergraph models.

Contribution

It offers the first precise threshold for reconstructing 3-uniform hypergraphs and bounds for higher uniformities, along with an efficient reconstruction algorithm and applications to hypergraph stochastic block models.

Findings

Exact threshold for d=3 hypergraph reconstruction

Bounds for d≥4 hypergraph reconstruction

Efficient algorithm for hypergraph recovery

Abstract

The graph projection of a hypergraph is a simple graph with the same vertex set and with an edge between each pair of vertices that appear in a hyperedge. We consider the problem of reconstructing a random $d$-uniform hypergraph from its projection. Feasibility of this task depends on $d$ and the density of hyperedges in the random hypergraph. For $d=3$ we precisely determine the threshold, while for $d\geq 4$ we give bounds. All of our feasibility results are obtained by exhibiting an efficient algorithm for reconstructing the original hypergraph, while infeasibility is information-theoretic. Our results also apply to mildly inhomogeneous random hypergrahps, including hypergraph stochastic block models (HSBM). A consequence of our results is an optimal HSBM recovery algorithm, improving on a result of Guadio and Joshi in 2023.