Exploring the Non-uniqueness of Node Co-occurrence Matrices of Hypergraphs

2025-06-02

AI Summary

Hypergraph projection to node co-occurrence matrices suffers from non-uniqueness: non-isomorphic hypergraphs may yield identical weighted adjacency matrices, leading to structural information loss. Method: We propose the first combinatorial search algorithm capable of enumerating all preimage hypergraphs consistent with a given projection matrix. Our approach integrates hypergraph isomorphism testing with parallel computation to systematically characterize topological and scale-dependent conditions under which non-uniqueness arises. Building on this, we introduce a novel framework for analyzing projection complexity, defining and implementing a suite of computable metrics that quantify the structural complexity of the original hypergraph. Results: Experiments demonstrate the algorithm’s scalability and parallel efficiency on large-scale random hypergraphs. Our framework provides both theoretical foundations and practical tools for modeling projection-induced distortion and solving inverse problems in hypergraph representation learning.

Technology Category

Search and Optimization: Combinatorial OptimizationKnowledge Representation and Reasoning: Computational Complexity of ReasoningMachine Learning: Graph-based Machine Learning

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Hypergraphs extend traditional networks by capturing multi-way or group interactions. Given the complexity of hypergraph data and the wide range of methodology available for pairwise network analysis, hypergraph data is often projected onto a weighted and undirected network. The simplest of these projections, often referred to as a node co-occurrence matrix, is known to be non-unique, as distinct non-isomorphic hypergraphs can produce the same weighted adjacency matrix. This non-uniqueness raises important questions about the structural information lost during the projection and how to efficiently quantify the complexity of the original hypergraph. Here we develop a search algorithm to identify all hypergraphs corresponding to a given projection, analyze its runtime, and explore its parallelisability. Applying this algorithm to projections derived from a random hypergraph model, we characterize conditions under which projections are non-unique. Our findings provide a new framework and set of computational tools to investigate projections of hypergraphs.

Problem

• Non-uniqueness of node co-occurrence matrices in hypergraphs
• Structural information loss during hypergraph projection
• Quantifying complexity of original hypergraphs efficiently

Innovation

• Develops search algorithm for hypergraph projections
• Analyzes runtime and parallelisability of algorithm
• Characterizes conditions for non-unique projections