Achievability of Heterogeneous Hypergraph Recovery from its Graph Projection

TL;DR

This paper investigates the conditions under which the original hyperedges of a heterogeneous hypergraph can be successfully reconstructed from its projected graph, extending known thresholds to more general cases.

Contribution

It introduces a new achievability result for hyperedge recovery in heterogeneous hypergraphs using clique-based algorithms, generalizing previous uniform hypergraph results.

Findings

Successful hyperedge recovery under certain density conditions

Generalization of known thresholds for uniform hypergraphs

Conjecture of optimality for the proposed threshold

Abstract

We formulate and analyze a heterogeneous random hypergraph model, and we provide an achieveability result for recovery of hyperedges from the observed projected graph. We observe a projected graph which combines random hyperedges across all degrees, where a projected edge appears if and only if both vertices appear in at least one hyperedge. Our goal is to reconstruct the original set of hyperedges of degree $d_j$ for some $j$. Our achievability result is based on the idea of selecting maximal cliques of size $d_j$ in the projected graph, and we show that this algorithm succeeds under a natural condition on the densities. This achievability condition generalizes a known threshold for $d$-uniform hypergraphs with noiseless and noisy projections. We conjecture the threshold to be optimal for recovering hyperedges with the largest degree.