Dynamical Processes on Metric Networks

Abstract

The structure of a network has a major effect on dynamical processes on that network. Many studies of the interplay between network structure and dynamics have focused on models of phenomena-such as disease spread, opinion formation and changes, coupled oscillators, and random walks-that one can describe using coupled ordinary differential equations, difference equations, stochastic processes, or agent-based models. In parallel to these developments, there have been many studies of wave propagation and other spatially extended processes on networks. These latter studies consider metric networks, in which the edges are associated with real intervals. Metric networks give a mathematical framework to describe dynamical processes that include both temporal and spatial evolution of some quantity of interest--such as the concentration of a diffusing substance or the amplitude of a wave--by using edge-specific intervals that quantify distance information between a network's nodes. Dynamical processes on metric networks often take the form of partial differential equations (PDEs). In this paper, we present a collection of techniques and paradigmatic linear PDEs that are insightful for analyzing the interplay between structure and dynamics in metric networks. We start by considering a time-independent Schro\"dinger equation. We then use both finite-difference and spectral approaches to study the Poisson, heat, and wave equations as paradigmatic examples of elliptic, parabolic, and hyperbolic PDEs on metric networks. In our numerical experiments, we consider metric networks with up to about 104 nodes and 104 edges. Additionally, using our spectral approach, we are able to resolve degenerate eigenmodes. A key contribution of our paper is to increase the accessibility of studying PDEs on metric networks. Software that implements our numerical approaches is available at https://gitlab.com/ComputationalScience/metric-networks.