# Dynamical processes on metric networks

CoRR（2024）

摘要

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. 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
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 useful to investigate the
interplay between structure and dynamics in metric networks. We start by
considering a time-independent Schrödinger 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 PDE
problems on metric networks. Our spectral approach is able to account for
degenerate eigenmodes. In our numerical experiments, we consider metric
networks with up to about 10^4 nodes and about 10^4 edges. 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.

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