Metrics in the Space of High Order Networks

TL;DR

This paper introduces new metrics for comparing high order networks, which are hypergraph-based structures capturing complex relationships, and demonstrates their effectiveness in analyzing coauthorship patterns.

Contribution

It develops and validates novel metrics for high order networks, extending traditional network comparison methods to hypergraph-based structures.

Findings

The metrics are valid and measure differences between high order networks.

They successfully distinguish collaboration patterns in coauthorship networks.

Properties of relationships between tuples of different lengths are established.

Abstract

This paper presents methods to compare high order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples. They can be considered as generalizations of conventional networks where only relationship functions between pairs are defined. Important properties between relationships of tuples of different lengths are established, particularly when relationships encode dissimilarities or proximities between nodes. Two families of distances are then introduced in the space of high order networks. The distances measure differences between networks. We prove that they are valid metrics in the spaces of high order dissimilarity and proximity networks modulo permutation isomorphisms. Practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers. The metrics succeed in identifying their respective collaboration patterns.