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 Schr\"odinger 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.

翻译：网络结构对其上的动力学过程具有重要影响。许多关于网络结构与动力学相互作用的研究聚焦于疾病传播、舆论形成与变化、耦合振子以及随机游走等现象的建模。与此同时，关于波传播及其他空间扩展过程在网络上的研究也大量涌现。这些后续研究考虑的是度规网络，其中边与实数区间相关联。度规网络通过使用量化节点间距离信息的边特定区间，为描述包含某种关注量（如扩散物质的浓度或波的振幅）时空演化的动力学过程提供了数学框架。度规网络上的动力学过程常以偏微分方程的形式呈现。在本文中，我们介绍了一系列技术和方法性线性偏微分方程，这些方程有助于研究度规网络中结构与动力学之间的相互作用。我们首先考虑时间无关的薛定谔方程，随后利用有限差分法和谱方法分别研究泊松方程、热方程和波动方程，作为度规网络上椭圆型、抛物型和双曲型偏微分方程问题的范例。我们的谱方法能够处理简并本征模。在数值实验中，我们考虑了节点数约10^4、边数约10^4的度规网络。本文的关键贡献在于提高了研究度规网络上偏微分方程的可及性。实现我们数值方法的软件可在https://gitlab.com/ComputationalScience/metric-networks获取。