Persistence of steady-states for dynamical systems on large networks

TL;DR

It is shown that if the graphon equation has a steady-state solution and if the linearization of the equation at that solution is invertible, then there exist related steady-state solutions to the finite-dimensional networked dynamical system over all sufficiently large graphs converging to the graphon.

Abstract

The goal of this work is to identify steady-state solutions to dynamical systems defined on large, random families of networks. We do so by passing to a continuum limit where the adjacency matrix is replaced by a non-local operator with kernel called a graphon. This graphon equation is often more amenable to analysis and provides a single equation to study instead of the infinitely many variations of networks that lead to the limit. Our work establishes a rigorous connection between steady-states of the continuum and network systems. Precisely, we show that if the graphon equation has a steady-state solution whose linearization is invertible, there exists related steady-state solutions to the finite-dimensional networked dynamical system over all sufficiently large graphs converging to the graphon. The proof involves setting up a Newton--Kantorovich type iteration scheme which is shown to be a contraction on a suitable metric space. Interestingly, we show that the first iterate of our defined operator in general fails to be a contraction mapping, but the second iterate is proven to contract on the space. We extend our results to show that linear stability properties further carry over from the graphon system to the graph dynamical system. Our results are applied to twisted states in a Kuramoto model of coupled oscillators, steady-states in a model of neuronal network activity, and a Lotka--Volterra model of ecological interaction.

Figures (7)

• Figure 1: (a) A cartoon of a 2-twisted state over a ring network with next-nearest-neighbor connections. (b) Contour plot of the small-world graphon \ref{['eq:IntroGraphon']} with $\alpha = 0.2$. (c) the pixel plot of the adjacency matrix of a random network built from the graphon according to \ref{['eq:IntroProb']}. Twisted states for the Kuramoto model of coupled oscillators in both the graphon (blue solid line) and random graph model (red dots) are provided with (d) $m = 2$, (e) $m = 3$, and (f) $m = 4$ twists.
• Figure 2: Pixel plots of Erdős-Rényi random graphs on $n = 10, 100,$ and $1000$ vertices generated by the graphon $W(x,y) = 1/2$ with black representing an edge and white representing no edge. Visually they appear to approach a solid gray state of value 1/2, which is the correct intuition for the cut norm in this case, but not the standard $L^p$ norms.
• Figure 3: A graph on 4 vertices (left) can be encoded as an adjacency matrix (center) which is used to define a step graphon (right). This step graphon representation is really a pixel plot of the adjaceny matrix, showing values of 0 in white and 1 in black.
• Figure 4: Homogeneous steady-states for the Wilson-Cowan type model (\ref{['continuous neuro']}). The left panel plots homogeneous steady-states for degree constant graphons as a function of the degree. Note the region of bistability. In the right panel we plot $\lambda/ (1+\mathrm{exp}(\mu-\delta u^*)$ and $\frac{u^*}{\Omega}$. Intersections of these curves represent steady-states and bistability is observed as $\Omega$ is varied.
• Figure 5: Left: Constant steady-state for (\ref{['continuous neuro']}) using $W(x,y) =p=0.5$ and $\lambda = \mu = \delta = 1$ gives $u^*\approx 0.15,$ shown in blue. Nearby steady-states for (\ref{['discrete neuro']}) with ER random graphs on 10 and 200 vertices shown in red. Right: Eigenvalues for $DF(u^*)$ using ER random graphs on $n$ vertices arrayed vertically for each $n$.

Theorems & Definitions (59)

• Lemma 2.1: lovasz12
• Lemma 2.2: vizuete2021laplacian
• Theorem 3.1
• Theorem 3.2
• Remark 1
• Lemma 4.1
• proof
• Corollary 4.2
• proof
• Lemma 4.3