Workshop on Metric Networks

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).

Over the last two to three decades, the study of metric networks has progressed in parallel to (and largely independent of) developments in more conventional network science. In network science, analysis of the interplay between network structure and dynamics has long been a key topic. Such research has used combinatorial networks, rather than metric networks, and accordingly it has focused primarily on ordinary differential equations and (both deterministic and stochastic) agent-based models on networks, rather than on PDEs.

This workshop provides a platform to connect researchers from the network science and metric networks communities who share a common interest in analytical, computational, and experimental studies of metric networks, such as quantum graphs, electrical networks, and gas networks.

Workshop dates: July 18 and 19, 2024

Workshop location: Room S0.03, Frankfurt School of Finance & Management (Adickesallee 32–34, 60322 Frankfurt am Main)

“Overview of Metric Networks” by Mason A. Porter (University of California, Los Angeles and Santa Fe Institute)

Abstract: I will overview metric networks and set the stage for our workshop. I will define metric networks and discuss their origin from quantum graphs. I will present both linear and nonlinear wave dynamics on  metric networks, and I will pose some thoughts that I hope that we can discuss throughout this workshop.

“Solving PDEs with Analog Computers” by Bernd Ulmann (FOM and Anabrid GmbH)

Abstract: Since digital computers are about to hit basic physical boundaries with respect to integration density, clock frequency, and energy efficiency, it is time to think about unconventional computing approaches for the solution of certain classes of problems. Particularly interesting are analog and hybrid computing techniques for the treatment of coupled DEQs and PDEs. This talk will outline three approaches for the solution of PDEs on analog and hybrid computers.

(This presentation will include live experiments!)

“Dynamical Processes on Metric Networks” by Lucas Böttcher (Frankfurt School and University of Florida)

Abstract: I will present a collection of techniques and paradigmatic linear PDEs that are useful to investigate the interplay between structure and dynamics in metric networks. I will start by considering a time-independent Schrödinger equation. I will 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.

“A Coupled Spatial-Network Model for Epidemiology” by Hannah Kravitz (Portland State University) and Christina Durón (Pepperdine University)

Abstract: There is extensive evidence that network structure (e.g., air transport, rivers, or roads) may significantly enhance the spread of epidemics into the surrounding geographical area, yet models coupling a network structure with a 2D area have largely not been studied. We discuss a model based on the Susceptible-Infected-Removed (SIR) system of differential equations which couples population centers, 1D travel routes, and a 2D continuum containing the bulk of the population. Some examples of implementation are provided and a discussion of the impact of the network structure on the spread of an epidemic follows.

“The Discrete Dirac Operator and the Mass of Simple and Higher-Order Networks” by Ginestra Bianconi (Queen Mary University of London)

Abstract: We discuss the properties of the discrete Dirac operator on simple and higher-order networks and its relevance to capture the coupled dynamics topological signals defined on nodes, links of graphs and even higher-dimensional simplices.
We will show how the Dirac operator can be coupled with the algebra of gamma matrices to define a Dirac field theory in which the spinor is given a geometrical interpretation. The field theory also includes metric degrees of freedom interpreted as the weights of the links for which we can define an action.  
We use the discrete topological Dirac operator to define an action for a massless self-interacting topological Dirac field inspired by the Nambu–Jona-Lasinio model. We propose a theoretical framework that explains how the mass of simple and higher-order networks emerges from their topology and geometry. The mass of the network is strictly speaking the mass of this topological Dirac field defined on the network; it results from the chiral symmetry breaking of the model and satisfies a self-consistent gap equation. Interestingly, it is shown that the mass of a network depends on its spectral properties, topology, and geometry. 

“Resonances in Quantum and Microwave Graphs” by Jiří Lipovský (University of Hradec Králové)

Abstract: We will introduce both theoretical and experimental results on resonances in quantum and microwave graphs. First, the definition and main properties of resonances will be introduced. Secondly, the high-energy asymptotics of the number of resonances will be studied and graphs with non-Weyl asymptotics will be shown. Finally, the experimental results that find the non-Weyl microwave graphs will be presented.

“Escape Times for Subgraph Detection and Graph Partitioning” by Jeremy L. Marzuola (University of North Carolina at Chapel Hill)

Abstract: We provide a rearrangement-based algorithm for fast detection of subgraphs of k vertices with long escape times for directed or undirected networks. Complementing other notions of densest subgraphs and graph cuts, our method is based on the mean hitting time required for a random walker to leave a designated set and hit the complement. We provide a new relaxation of this notion of hitting time on a given subgraph and use that relaxation to construct a fast subgraph detection algorithm and a generalization to K-partitioning schemes. Using a modification of the subgraph detector on each component, we propose a graph partitioner that identifies regions where random walks live for comparably large times.  A version of this can be implemented in continuum domains and in metric graphs. I’ll discuss the original results and extensions we are working on currently.

“Efficient Transient Simulation and Optimization of Large-Scale Gas Networks” by Pia Domschke (Frankfurt School)

Abstract: The operation of gas transmission networks involves many challenges for the network operators in the real market. Fed in by multiple suppliers, gas has to be routed through the network to meet the consumers’ demands. At the same time, the operational costs of the network like energy consumption of compressor stations or contractual penalties have to be minimized. This leads to an optimal control problem on a network.
For the optimization task, reliable simulation results are necessary. This can be achieved by using a goal-oriented adaptive strategy for the simulation, which ensures the desired accuracy while reducing computation time. Besides an adaptive spatial discretization and variable time stepping, simplified physical models are used in regions of the network with low activity, while sophisticated models are applied in regions where the dynamical behaviour of the flow needs to be resolved in detail. The adjoint equations computed for the a posteriori error estimators used here can also be used in a first-discretize-then-optimize strategy to achieve a desired state of the gas network, e.g. in nomination validation.
We will present the adaptive simulation algorithm and show some numerical experiments as well as the applicability in an optimization framework.

"Spectral solutions of PDEs on Networks" by Jean-Guy Caputo (INSA Rouen Normandie)

Abstract: Partial differential equations on networks occur in many applications such as fluid networks, electromechanical waves on a transmission grid, etc.. The mathematical model consists in a metric graph where vertices are connected by arcs bearing a metric. We consider standard coupling conditions at the vertices: continuity of the field and zero total gradient (Kirchoff's law).
The standard 1D Laplacian together with these boundary conditions yields a complete orthogonal basis of the L2 space extended to all arcs. With this spectral framework, linear partial differential equations on the metric graph can be treated as in 1D on a finite interval. 
For this generalized Laplacian, we introduce a systematic procedure to compute the eigenvalues/eigenfunctions of arbitrary order for a general metric graph. These will be used to describe solutions of the heat and wave equation.
Regarding methodology, we estimate the number of modes necessary to get the required precision. We also show that the method is spectrally accurate for initial conditions that have compact support on all arcs. Otherwise the decay of the spectral coefficients a_q is like 1/q^₄ .
We also suggest practical ways to design resonating systems based on metric graphs. Finally, numerical simulations of the time-dependent wave equation on the metric graph show that localized eigenvectors can be excited by a broadband initial condition, even with leaky boundary conditions.
This is joint work with Moysey Brio and Hannah Kravitz.

"The Duistermaat Index and Eigenvalue Interlacing for Graphs with General Self-Adjoint Conditions" by Gregory Berkolaiko (Texas A&M University)

Abstract: Eigenvalue interlacing is a tremendously useful tool in linear algebra and spectral analysis.  In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts the eigenvalue up, but not further than the next unperturbed eigenvalue.  For different types of perturbations, this idea is known as the "Weyl interlacing" (additive perturbations), "Cauchy interlacing" (for principal submatrices of Hermitian matrices), "Dirichlet-Neumann bracketing" and so on.
We discuss the extension of this idea to general perturbations in vertex conditions of a quantum graph.  In this context, even the terms such as "signature of the perturbation" are not immediately clear, since one cannot take the difference of two operators with different domains.  However, it turns out that definitive answers can be obtained, and they are expressed most concisely terms of the Duistermaat index, an integer-valued topological invariant describing the relative position of three Lagrangian planes in a symplectic space.  Two of the Lagrangian planes describe the two sets of vertex conditions, while the third one corresponds to the distinguished Friedrichs condition (which in many cases takes the form of the Dirichlet condition on the affected vertices).
We will illustrate our general results with simple examples, avoiding technicalities as much as possible and giving intuitive explanations of the Duistermaat index, the rank and signature of the perturbation in the self-adjoint extension, and the curious role of the third extension (Friedrichs) appearing in the answers.
Based on a joint work with Graham Cox, Yuri Latushkin and Selim Sukhtaiev.

"Discovering Hidden Layers in Quantum Graphs" by Julian Sienkiewicz (Warsaw University of Technology)

Abstract: Finding hidden layers in complex networks is an important and a nontrivial problem in modern science. We explore the framework of quantum graphs to determine whether concealed parts of a multilayer system exist and if so then what is their extent, i.e., how many unknown layers are there. Assuming that the only information available is the time evolution of a wave propagation on a single layer of a network it is indeed possible to uncover that which is hidden by merely observing the dynamics. We present evidence on both synthetic and real-world networks that the frequency spectrum of the wave dynamics can express distinct features in the form of additional frequency peaks. These peaks exhibit dependence on the number of layers taking part in the propagation and thus allowing for the extraction of said number. We show that, in fact, with sufficient observation time, one can fully reconstruct the row-normalized adjacency matrix spectrum. We compare our propositions to a machine learning approach using a wave packet signature method modified for the purposes of multilayer systems.
Based on a joint work with Łukasz Gajewski and Janusz Hołyst.

"Quantum Chaos in Quantum Graphs: Theory and Experiment" by Barbara Dietz-Pilatus (Institute for Basic Science)

It has been established that the spectral properties of generic quantum systems with chaotic classical counterparts are universal. According to the Bohigas--Giannoni--Schmit conjecture, they coincide for systems with orthogonal, unitary, or symplectic symmetry with those of random matrices from the Gaussian orthogonal (GOE), unitary (GUE) or symplectic (GSE) ensembles. In quantum systems with unitary symmetry, time-reversal (T) invariance is violated. Systems belonging to the orthogonal or the symplectic universality class correspond to integer and half-integer spin systems, respectively, with pre-served T invariance. Quantum graphs provide a particularly suitable model to emulate such quantum systems, especially because all three universality classes can be realized experimentally with microwave networks. I will report on the theoretical and experimental investigation of the spectral properties of closed quantum graphs, and of the scattering matrix of open ones employing quantum graphs. Finally, I will speak about recent experiments with waveguide networks as model for quantum graphs.

Please register for the free workshop using the registration button.