Dimensional reduction of dynamical systems on graphons

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.