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

TL;DR

This paper demonstrates that certain nonlinear graph dynamics can be exactly or approximately represented as linear dynamics on higher-order structures like hypergraphs, revealing a duality between structure and dynamics.

Contribution

It introduces a duality framework showing how nonlinear graph dynamics can be transformed into linear dynamics on higher-order networks, including exact and approximate representations.

Findings

Finite polynomial dynamics are exactly linearizable on hb-graphs of the same size.

Exact linear representation for general nonlinearities requires infinite hb-graphs.

Finite truncations of infinite hb-graphs approximate the original nonlinear dynamics.

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 work 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. Using Carleman linearisation arguments, we show that finite polynomial dynamics defined in the $|V|$ vertices of a graph admit an exact representation as linear dynamics on the state space of an hb-graph of order $|V|$, a combinatorial structure that extends hypergraphs by allowing vertex multiplicity, where the specific shape of the nonlinearity indicates whether the hb-graph is either finite or infinite (in terms of the number of hb-edges). For more general analytic nonlinearities, exact linear representation always require an hb-graph of infinite size, and its finite-size truncation provides an approximate representation of the original nonlinear graph-based dynamics.