Dimensional reduction of dynamical systems on graphons

TL;DR

This work develops a graphon-based continuum framework to reduce high-dimensional dynamical systems on networks to one-dimensional models. It proves the existence, uniqueness, and convergence of graphon dynamics for dense networks and introduces two dimension-reduction schemes, the GBB and spectral reductions, with a Galerkin approximation linking finite networks to the graphon limit. Through numerical experiments on six dynamical models and six graphons, the reductions prove accurate in many settings (notably ER, ring, and small-world graphons) and demonstrate the practical potential for scalable analysis of large networks. The study also outlines limitations, including incomplete convergence proofs for the spectral reduction and the focus on dense graphons, suggesting avenues for extending the theory to sparse graphs and broader graphon classes.

Abstract

Dynamical systems on networks are inherently high-dimensional unless the number of nodes is extremely small. Dimension reduction methods for dynamical systems on networks aim to find a substantially lower-dimensional system that preserves key properties of the original dynamics such as bifurcation structure. A class of such methods proposed in network science research entails finding a one- (or low-) dimensional system that a particular weighted average of the state variables of all nodes in the network approximately obeys. We formulate and mathematically analyze this dimension reduction technique for dynamical systems on dense graphons, or the limiting, infinite-dimensional object of a sequence of graphs with an increasing number of nodes. We first theoretically justify the continuum limit for a nonlinear dynamical system of our interest, and the existence and uniqueness of the solution of graphon dynamical systems. We then derive the reduced one-dimensional system on graphons and prove its convergence properties. Finally, we perform numerical simulations for various graphons and dynamical system models to assess the accuracy of the one-dimensional approximation.

Figures (8)

• Figure 1: Visualization of the graphons used in the present study. (a) ER graphon with $p = 0.1$. (b) Ring graphon with $q=1/3$. (c) Small-world graphon with $p = 0.1$ and $q = 1/3$. (d) Power-law graphon with $C= 0.5$ and $\nu = -0.2$. (e) Modular graphon with $\gamma = 1/3$, $p_{\rm in} = 0.2$, and $p_{\rm out} = 0.01$. (f) Bipartite graphon with $\gamma = 1/3$ and $p = 0.1$.
• Figure 2: Accuracy of the two dimension reduction methods for the SIS dynamics. (a) ER graphon, GBB reduction. (b) ER graphon, spectral reduction. (c) Ring, GBB. (d) Ring, spectral. (e) Small-world, GBB. (f) Small-world, spectral. (g) Power-law, GBB. (h) Power-law, spectral. (i) Modular, GBB. (j) Modular, spectral. (k) Bipartite, GBB. (l) Bipartite, spectral. The thick semi-transparent red lines represent the numerically obtained one-dimensional observable obtained from simulations of the full graphon dynamical system. The thin black lines represent the equilibria of the GBB or spectral reduction.
• Figure 3: Accuracy of the two dimension reduction models for the coupled double-well dynamics. (a) ER, GBB. (b) ER, spectral. (c) Ring, GBB (d) Ring, spectral. (e) Small-world, GBB. (f) Small-world, spectral. (g) Power-law, GBB. (h) Power-law, spectral. (i) Modular, GBB. (j) Modular, spectral. (k) Bipartite, GBB. (l) Bipartite, spectral. The thick semi-transparent red and blue lines represent the numerically obtained one-dimensional solutions when the initial states are lower and upper, respectively. The thin black lines represent the equilibria of the GBB or spectral reductions with the lower and upper initial conditions altogether.
• Figure 4: Accuracy of the two dimension reduction models for the gene-regulatory dynamics. See the caption of Fig. \ref{['fig_DW_Model']} for the legends.
• Figure 5: Accuracy of the two dimension reduction models for the GLV dynamics. See the caption of Fig. \ref{['fig_SIS_Model']} for the legends.

Theorems & Definitions (13)

• Theorem 4: Existence and uniqueness of the solution
• proof : Proof of Theorem \ref{['theorem_1']}
• Theorem 5: Boundedness of the solution
• proof : Proof of Theorem \ref{['theorem_2']}
• Lemma 1
• Theorem 6
• Definition 1
• Theorem 7
• proof
• Theorem 8