Partial and Exact Recovery of a Random Hypergraph from its Graph Projection

TL;DR

This paper investigates the conditions under which a random hypergraph can be exactly or partially reconstructed from its graph projection, resolving previous conjectures and establishing thresholds for recovery fidelity.

Contribution

It provides a complete characterization of hypergraph recovery from projections for all degrees and recovery types, including the impact of edge multiplicity information.

Findings

Exact recovery is possible below a certain degree threshold.

Recovery fidelity exhibits an all-or-nothing phase transition.

The work resolves all conjectures from prior related research.

Abstract

Consider a $d$-uniform random hypergraph on $n$ vertices in which hyperedges are included iid so that the average degree is $n^\delta$. The projection of a hypergraph is a graph on the same $n$ vertices where an edge connects two vertices if and only if they belong to some hyperedge. The goal is to reconstruct the hypergraph given its projection. An earlier work of Bresler, Guo, and Polyanskiy (COLT 2024) showed that exact recovery for $d=3$ is possible if and only if $\delta < 2/5$. This work completely resolves the question for all values of $d$ for both exact and partial recovery and for both cases of whether multiplicity information about each edge is available or not. In addition, we show that the reconstruction fidelity undergoes an all-or-nothing transition at a threshold. In particular, this resolves all conjectures from Bresler, Guo, and Polyanskiy (COLT 2024).