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

TL;DR

This work resolves the problem of reconstructing a random $d$-uniform hypergraph from its graph projection, providing sharp thresholds for both partial and exact recovery across all $d$ and for weighted versus unweighted projections. The authors develop a framework combining a maximum clique cover approach, preimage-overlap analysis, and information-theoretic and combinatorial arguments to prove all-or-nothing phase transitions and characterize exact-recovery bottlenecks via ambiguous graphs. The exact-recovery threshold equals the minimum of $\frac{d-1}{d+1}$ and $\frac{2d-4}{2d-1}$ (which is $\frac{d-1}{d+1}$ for $d\ge 5$), while partial recovery undergoes a transition at $\delta=(d-1)/(d+1)$; when using weighted projections, the partial and exact thresholds align at $\delta=(d-1)/(d+1)$, reflecting altered bottlenecks. Overall, the paper completes the recovery landscape for projections of random hypergraphs and demonstrates all-or-nothing behavior, with implications for planted CSPs and inverse problems in higher-order network data.

Abstract

Consider a $d$-uniform random hypergraph on $n$ vertices in which hyperedges are included iid so that the average degree is $n^δ$. 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 $δ< 2/5$. 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).

Figures (1)

• Figure 1: This figure displays $G_{a,3}$. One minimal preimage consists of the gray and red triangles while the other consists of the gray and blue triangles. For $d>3$ the red and blue triangles are replaced with cliques of size $d$, the sets $S_i^u$, $S_i^w$ change to $(d-2)$-cliques and their number changes from 2 to $d-1$.

Theorems & Definitions (96)

• Theorem 1.1: Partial Recovery
• Theorem 1.2: Partial Recovery for Weighted Projection
• Theorem 1.3: Exact Recovery
• Theorem 1.4: Exact Recovery for Weighted Projection
• Remark 1.5
• Definition 1.6
• Definition 1.7
• Lemma 2.1
• Theorem 2.2
• Corollary 2.3