Measuring dynamical systems on directed hyper-graphs

TL;DR

This paper explores how to analyze dynamical systems like random walks on directed hyper-graphs, extending network measures to higher-order interactions without excessive computational complexity.

Contribution

It introduces a method to measure dynamical systems on directed hyper-graphs by leveraging existing network measures on the associated transition matrix.

Findings

Established a framework for applying network measures to hypergraph-based dynamical systems.

Demonstrated that the approach simplifies analysis of higher-order interactions.

Provided insights into the structure-dynamics relationship in complex systems.

Abstract

Networks and graphs provide a simple but effective model to a vast set of systems which building blocks interact throughout pairwise interactions. Unfortunately, such models fail to describe all those systems which building blocks interact at a higher order. Higher order graphs provide us the right tools for the task, but introduce a higher computing complexity due to the interaction order. In this paper we analyze the interplay between the structure of a directed hypergraph and a linear dynamical system, a random walk, defined on it. How can one extend network measures, such as centrality or modularity, to this framework? Instead of redefining network measures through the hypergraph framework, with the consequent complexity boost, we will measure the dynamical system associated to it. This approach let us apply known measures to pairwise structures, such as the transition matrix, and determine a family of measures that are amenable of such procedure.