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

2025-02-20

AI Summary

This work addresses the problem of recovering a $d$-uniform random hypergraph from its graph projection, unifying the precise characterization of exact and partial recovery phase transitions for all $d geq 2$. Methodologically, it integrates information-theoretic analysis, random hypergraph modeling, and statistical inference, leveraging first- and second-moment methods together with refined likelihood ratio analysis. The paper establishes, for the first time, an explicit critical threshold $delta_c(d)$: exact recovery is information-theoretically feasible with high probability if and only if the sampling density $delta < delta_c(d)$; otherwise, it is impossible. Partial recovery exhibits a sharp phase transition and rigorously confirms the “all-or-nothing” phenomenon. These results fully resolve all conjectures posed at COLT 2024, covering both settings—with and without edge multiplicity information—and establish the first complete phase transition theory for hypergraph structure inference.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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).

Problem

• Reconstruct hypergraph from its graph projection.
• Determine recovery thresholds for various hypergraph parameters.
• Analyze all-or-nothing transition in reconstruction fidelity.

Innovation

• Hypergraph reconstruction from projection
• Threshold-based fidelity transition
• Exact and partial recovery analysis