Dynamical Systems on Graph Limits and Their Symmetries

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.

Figures (8)

• Figure 1: This graphon has a block structure which is a canonical embedding on the unit interval of a finite graph with $5$ vertices.
• Figure 2: Infinite binary tree.
• Figure 3: The connection strength between a vertex $x\in [0,1/5]$ and a vertex $y \in (1/5,1]$ depends on $y$ but not on $x$.
• Figure 4: Representing a naturally $2$-dimensional index space on the $1$-dimensional unit interval can create complicated, fractal-like pictures, which hide the underlying graphon symmetries. The graphon in this figure has automorphism group $\mathrm D_4 \ltimes \mathbb T^2$.
• Figure 5: We simulate the evolution of an initial condition with symmetry $(x,y)\mapsto(-y,x)$ with respect to the coupling function $g(u_x,u_y) = \sin(u_y-u_x + 1)$. The $2$-dimensional torus is approximated by the graph $C_{30}\times C_{30}$. In the picture, vertices are arranged in a square for visual purposes: vertices on opposite sides are understood to be connected. The pattern is preserved over time.

Theorems & Definitions (18)

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• proof : Proof of Theorem \ref{['thm:twins_constant']} in the case $J=I$
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