Modular dynamical systems on networks

TL;DR

This paper introduces a novel framework for analyzing continuous-time dynamical systems on networks using graph fibrations, enabling the derivation of conjugacies, synchrony subspaces, and system abstractions.

Contribution

It establishes a new connection between graph fibrations and dynamical system mappings, providing tools for system reduction and synchronization analysis.

Findings

Graph fibrations induce conjugacies between network dynamical systems.

Surjective fibrations lead to synchrony subspaces in networks.

Injective fibrations enable system reduction through surjective maps.

Abstract

We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller ones. One can view these surjections as a kind of "fast/slow" variable decompositions or as "abstractions" in the computer science sense of the word.