Beyond Pairwise Relationships: Modeling Real-world Dynamics Via High-order Networks

Abstract

Every single person, animal, or thing we can see in the world around us is part of a broader collection of components that can spontaneously self- organize to exhibit non-trivial global structures and behaviors at larger

scales, often without external intervention, central authorities, or lead- ers. The properties of the collection these components give life cannot be

understood or predicted from the full knowledge of its elements alone.

Each collection is an example of complex systems whose behavior is in- trinsically challenging to model due to the high non-linearity of the inter- actions between its constituents.

Traditionally, complex systems have been successfully studied through

graphs abstracting the underlying network with vertices and edges con- necting pairs of interacting components. Over the years, the scientifc

community has enriched the graph modeling framework for better cap- turing the richness of the interactions among such units. However, graphs

have a substantial limitation encoded in their nature: they exclusively cap- ture pairwise interactions. Yet, many complex systems are characterized

by group interactions that cannot be described simply in terms of dyads. Studying such systems hence require new mathematical frameworks and scientifc methodologies for its investigation.

Hypergraphs are the perfect candidates to tackle this task. A hyper- graph is a generalization of a graph, where a (hyper)edge allows the con- nection of an arbitrary number of vertices. However, the powerful ex- pressiveness of hypergraphs has a few drawbacks: dealing with the com- plexity of such data structures and the lack of appropriate tools and al- gorithms for their study. For this reason, hypergraphs have been little

used in literature in favor of their graph-counterpart. Recently, this trend

has been drifting, thanks to an increasing number of systematic stud- ies demonstrating that considering the higher-order structure of complex

systems can enhance our modeling capacities and help us understand and predict their dynamical behavior. [...] [edited by Author]