Distances in Higher-Order Networks and the Metric Structure of Hypergraphs

Area of Science:

• Network Science
• Graph Theory
• Data Analysis

Background:

• Traditional network analysis often relies on pairwise interactions, limiting the study of complex systems.
• Higher-order interactions, represented by hypergraphs, are crucial in many real-world phenomena but are challenging to analyze quantitatively.
• Existing network metrics may not adequately capture the structural properties of hypergraphs.

Purpose of the Study:

• To develop a novel metric for quantifying distances in hypergraphs.
• To extend classic network centrality measures (efficiency, closeness, betweenness) to hypergraphs.
• To reveal new insights into the structural features and node roles in complex networks with higher-order interactions.

Main Methods:

• Introduced a new hypergraph distance metric considering inter-node distances within hyperedges and distances between hyperedges.
• Utilized computations on a weighted line graph of the hypergraph.
• Applied the metric to synthetic and large-scale real-world hypergraphs.
• Generalized efficiency, closeness, and betweenness centrality for hypergraphs.

Main Results:

• The novel metric effectively unveils structural information in synthetic hypergraphs.
• Analysis of real-world hypergraphs using the new metric provides insights beyond pairwise interaction analysis.
• Generalized centrality measures offer different node assessments compared to hypergraph clique projections, especially for hypergraphs with large hyperedges.

Conclusions:

• The proposed hypergraph distance metric is a valuable tool for analyzing complex networks with higher-order interactions.
• The generalized centrality measures provide a more nuanced understanding of node roles and information transferability in hypergraphs.
• This approach enhances the study of network structures beyond traditional pairwise relationships.