On the equivalence between nonlinear graph-based dynamics and linear dynamics on higher-order networks

TL;DR

This short note investigates how and when a nonlinear dynamics defined on the vertex set of a graph allows an equivalent representation in terms of a linear dynamics defined on the state space of a sufficiently richer, higher-order interaction structure, and shows that multilinear dynamics defined in the vertices of a graph admit an exact finite realizations as linear dynamics on the state space of a hypergraph.

Abstract

In network science, collective dynamics of complex systems are typically modelled as (nonlinear, often including many-body) vertex-level update rules evolving over a graph interaction structure. In recent years, frameworks that explicitly model such higher-order interactions in the interaction backbone (i.e. hypergraphs) have been advanced, somehow shifting the imputation of the effective nonlinearity from the dynamics to the interaction structure. In this short note we discuss such structural-dynamical representation duality, and investigate how and when a nonlinear dynamics defined on the vertex set of a graph allows an equivalent representation in terms of a linear dynamics defined on the state space of a sufficiently richer, higher-order interaction structure. We show that multilinear dynamics defined in the vertices of a graph admit an exact finite realizations as linear dynamics on the state space of a hypergraph. For other high-order interactions involving more general analytic nonlinearities, using Carleman linearization theory we discuss how that the required state space liftings necessary to linearize the dynamics cannot be accomodated to the simple structure of a hypergraph, and a richer combinatorial architecture such as a hb-graph is needed.