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

TL;DR

This paper investigates the non-uniqueness of node co-occurrence matrices derived from hypergraphs, developing algorithms to identify all hypergraphs corresponding to a given projection and analyzing the conditions for non-uniqueness.

Contribution

It introduces a search algorithm to find all hypergraphs matching a projection and explores the conditions leading to non-uniqueness, advancing hypergraph analysis methods.

Findings

Identifies conditions for non-uniqueness of hypergraph projections

Develops an efficient search algorithm for hypergraph reconstruction

Analyzes the runtime and parallelisability of the algorithm

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.