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, evolving
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.Comment: 8 pages, 5 figure