Thresholds for Reconstruction of Random Hypergraphs From Graph Projections

2025-02-12

Annual Conference Computational Learning Theory

AI Summary

This paper investigates the feasibility of reconstructing a random $d$-uniform hypergraph from its graph projection—the union of complete subgraphs induced by each hyperedge. Using probabilistic analysis, information-theoretic arguments, and combinatorial reasoning, we establish the exact information-theoretic phase transition threshold for $d=3$, and derive tight, matching upper and lower bounds on the reconstruction threshold for $d geq 4$. We further propose a polynomial-time reconstruction algorithm achieving the information-theoretic optimum. Extending our framework to mildly non-uniform settings—including the hypergraph stochastic block model (HSBM)—we obtain the first information-theoretically optimal community recovery algorithm, significantly improving upon the prior work of Guadio–Joshi (2023). Our results unify and advance the theoretical understanding of hypergraph learning from pairwise projections, bridging gaps between statistical identifiability and efficient computability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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
• Web Mining and Content Analysis: Models for Web evolution

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 $dgeq 4$ 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.

Problem

• Reconstruct random d-uniform hypergraphs
• Determine thresholds for hypergraph reconstruction
• Improve HSBM recovery algorithm efficiency

Innovation

• Efficient hypergraph reconstruction algorithm
• Threshold determination for hypergraph feasibility
• Optimal HSBM recovery algorithm improvement