Dynamical Systems on Graph Limits and Their Symmetries

TL;DR

This paper explores how symmetries in graph limits like graphons influence the dynamics of interacting units, revealing new types of symmetries and their effects on large networks.

Contribution

It characterizes symmetry properties of dynamical systems on graph limits, including generalized noninvertible symmetries, and links these to network dynamics.

Findings

Symmetries in graph limits shape the dynamics and invariant subspaces.

Large asymmetric networks can exhibit symmetric behaviors due to their limits.

Generalized noninvertible symmetries are supported in dynamics on graph limits.

Abstract

The collective dynamics of interacting dynamical units on a network crucially depends on the properties of the network structure. Rather than considering large but finite graphs to capture the network, one often resorts to graph limits and the dynamics thereon. We elucidate the symmetry properties of dynamical systems on graph limits -- including graphons and graphops -- and analyze how the symmetry shape the dynamics, for example through invariant subspaces. In addition to traditional symmetries, dynamics on graph limits can support generalized noninvertible symmetries. Moreover, as asymmetric networks can have symmetric limits, we note that one can expect to see ghosts of symmetries in the dynamics of large asymmetric networks.