Dynamical processes on metric networks

TL;DR

This work studies dynamical processes on metric networks by formulating PDEs on edge intervals and employing self-adjoint operators like the generalized Laplacian $\tilde{\Delta}$. It develops a robust spectral method that accounts for degenerate eigenmodes and couples it with finite-difference approaches to solve the Poisson, heat, and wave equations on networks of up to $\sim 10^4$ edges/nodes, with open-source code available at $\url{https://gitlab.com/ComputationalScience/metric-networks}$. Key contributions include explicit handling of Kirchhoff/Dirichlet coupling, Weyl-law-based mode counting, and detailed demonstrations on star and lattice networks, illustrating accurate spectral expansions and time-evolution dynamics. The results enable scalable analysis of spatially extended dynamics on networks, with potential applications in physics, engineering, and network science, and provide a practical toolkit for researchers to simulate PDEs on metric graphs.

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. 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.

Figures (12)

• Figure 1: An example of a metric network with $N = 8$ nodes and $M = 10$ edges. The length of an edge that connects two nodes $u$ and $v$ is $\ell_{uv}$. The arrows indicate the starting and ending points of an interval. For example, the edge that connects nodes 1 and 2 starts at node 1 and ends at node 2. The depicted metric network is embedded in $\mathbb{R}^2$, so each node is associated with a location in the plane. The blue regions indicate polygons whose vertices correspond to the network nodes. The edges of these polygons correspond to the intervals (but without their directions).
• Figure 1: A finite-difference discretization of edge $i$ with length $\ell_i$. We employ a uniform discretization with step size $h_i = \ell_i/N_i$.
• Figure 1: Eigenmodes of the Schrödinger equation \ref{['eq:helmholtz']} on a metric star network with $N = 4$ nodes (black disks) and $M = 3$ edges (black lines). All edges have length $\ell$ and are equipped with Kirchhoff boundaries. For $m \in \{1,3\}$, the eigenmodes are degenerate [see Eq. \ref{['eq:star_degenerate']}]; they are not degenerate for $m \in \{2,4\}$ [see Eq. \ref{['eq:star_not_degenerate']}]. The solid blue curves show the analytical solutions \ref{['eq:star_degenerate']} and \ref{['eq:star_not_degenerate']}. We also determine the characteristic wavenumbers $k_m$ numerically by searching for minima of the inverse condition number of the coupling-condition matrix $T_{ }(k)$ [see Eq. \ref{['eq:T_k_star_graph']}]. We then use these numerical $k_m$ to compute the nullspace (and hence the eigenmodes) of the coupling-condition matrix using a QR decomposition. We indicate the associated numerical eigenmodes with dashed red curves.
• Figure 1: Solution of the Poisson equation on a metric star network with $N = 4$ nodes (black dots) and $M = 3$ edges (black lines). All edges $i \in \{1,2,3\}$ have length $\ell_i = 1$ and Kirchhoff boundaries. (a) The source term $\rho_i(x_i) = \cos(2\pi x_i)$ (solid blue curves) on the right-hand side of the Poisson equation \ref{['eq:poisson']}. (b) The solution $\phi_i(x_i) = -\cos(2\pi x_i)/(4\pi^2)$ (solid blue curves) of the Poisson equation. The dashed red and dotted orange curves indicate numerical solutions using finite-difference and spectral methods, respectively.
• Figure 1: Computation time for solving the Poisson equation \ref{['eq:poisson']} on metric hexagonal lattices with a finite-difference approach for different numbers $N$ of nodes and numbers $M$ of edges: $(N,M) \in \{(106,145),(202,281),(394,553),(778,1097),(1546,2185),(3082,4361),(6154,8713)\}$. All edges $i \in \{1,\ldots,M\}$ have length $\ell_i = 1$ and Kirchhoff boundaries. The source term on the right-hand side of the Poisson equation \ref{['eq:poisson']} is $\rho_i(x_i) = \cos(2\pi x_i)$ for each edge. The solution of the Poisson equation is $\phi_i(x_i) = -\cos(2\pi x_i)/(4\pi^2)$ for each edge. The dashed black line corresponds to a power law with exponent $2$.

Theorems & Definitions (1)

• Lemma A.1: Schur's Lemma