Measuring dynamical systems on directed hypergraphs

Networks and graphs provide a simple but effective model to a vast set of systems in which building blocks interact throughout pairwise interactions. Unfortunately, such models fail to describe all those systems in 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 to such a procedure.