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## Über dieses Buch

Dynamical models on graphs or random graphs are increasingly used in applied sciences as mathematical tools to study complex systems whose exact structure is too complicated to be known in detail. Besides its importance in applied sciences, the field is increasingly attracting the interest of mathematicians and theoretical physicists also because of the fundamental phenomena (synchronization, phase transitions etc.) that can be studied in the relatively simple framework of dynamical models of random graphs. This volume was developed from the Mathematical Technology of Networks conference held in Bielefeld, Germany in December 2013. The conference was designed to bring together functional analysts, mathematical physicists, and experts in dynamical systems. The contributors to this volume explore the interplay between theoretical and applied aspects of discrete and continuous graphs. Their work helps to close the gap between different avenues of research on graphs, including metric graphs and ramified structures.

## Inhaltsverzeichnis

### Lack of Ground State for NLSE on Bridge-Type Graphs

We prove the nonexistence of ground states for NLSE on bridge-like graphs, i.e. graphs with two halflines and four vertices, of which two at infinity, with Kirchhoff matching conditions. By ground state we mean any minimizer of the energy functional among all functions with the same mass.
Riccardo Adami, Enrico Serra, Paolo Tilli

### Instability of Stationary Solutions of Evolution Equations on Graphs Under Dynamical Node Transition

The nonexistence of stable stationary nonconstant solutions of reaction–diffusion-equations $$\partial _{t}u_{j} = \partial _{j}\left (a_{j}(x_{j})\,\partial _{j}u_{j}\right ) + f(u_{j})$$ on the edges of a finite metric graph is investigated under continuity and dynamical consistent Kirchhoff flow conditions at all vertices v i of the graph:
$$\displaystyle{\sum _{j}d_{\mathit{ij}}a_{j}(v_{i})\partial _{j}u_{j}(v_{i}) +\sigma _{i}\partial _{t}u(v_{i}) = 0.}$$
Various instability criteria are presented, in particular, for some classes of polynomial reaction terms f.
Joachim von Below, Baptiste Vasseur

### Statistical Characterization of a Small World Network Applied to Forest Fires

The characteristics of the propagation of forest fires under the influence of firebrands and the interaction zone (lx, ly) due to radiation are examined using the model of Small World Network. We analyze the distribution of the connections in a Small World Network and the cluster coefficient that represent the mathematical properties of the network. The used model is a stochastic model for predicting the behavior of wildfires. It is a variant of the social Small World Network, initially proposed by Watts and Strogatz, which allows the creation of more clusters and connections over long distances. This model was successfully applied to the spread of diseases and is characterized by a strong performance in clusters and a Poisson distribution of connections. The Model of Small World Network has also been adapted to study the spread of forest fires where it can include connections beyond nearest neighbors due to radiation from the flames or fire surges induced by firebrands which other propagation models cannot. It has been validated by experimental results of real fires. The main goal of this paper consists to investigate the most robust measures of network topology for a heterogeneous and∕or homogeneous system near the percolation threshold.
Fatima Zahra Benzahra Belkacem, Noureddine Zekri, Mekki Terbeche

### Network Dynamics as an Inverse Problem

Power grids, transportation systems, neural circuits and gene regulatory networks are just some of the many examples of networks in action. To understand mechanisms underlying collective network dynamics, typically a forward perspective is taken and mathematical models of given systems are explored as a function of their parameters. One question, for instance, might be how the collective dynamics undergoes a bifurcation when the network connectivity is changed. Here, we propose an inverse perspective on. We determine, based on the units’ time series, the set of all networks that generate a given collective dynamics. In particular, we show how the dynamics of a network may be parametrized in the phase portrait. Interestingly, even networks with very different connection topologies may generate identical dynamics. As an example, we rewire networks of Kuramoto-like oscillators with random network topologies into different networks that display the same collective time evolution. The results offer an alternative view on studying the interplay between the structure and dynamics of complex networks.

### Dynamics on a Graph as the Limit of the Dynamics on a “Fat Graph”

We discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (“fat graph”) when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph. Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.
Gianfausto Dell’Antonio, Alessandro Michelangeli

### Spectral Inequalities for Quantum Graphs

We review our joint work with Evans Harrell on semiclassical and universal inequalities for quantum graphs. The proofs of these inequalities are based on an abstract trace inequality for commutators of operators. In this article we give a new proof of this abstract trace inequality. Another ingredient in proving semiclassical and universal inequalities is an appropriate choice of operators in this trace inequality. We provide a new approximation method for such a choice.
Semra Demirel-Frank

### Intrinsic Metrics on Graphs: A Survey

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we determine the spectrum as a set via solutions, discuss upper and lower spectral bounds by isoperimetric constants and volume growth and study p-independence of spectra under a volume growth assumption.
Matthias Keller

### Spectral Gap for Complete Graphs: Upper and Lower Estimates

Lower and upper estimates for the spectral of the Laplacian on a compact metric graph are discussed. New upper estimates are presented and existing lower estimates are reviewed. The accuracy of these estimates is checked in the case of complete (not necessarily regular) graph with large number of vertices.
Pavel Kurasov

### Sharp Spectral Estimates for Periodic Matrix-Valued Jacobi Operators

For the periodic matrix-valued Jacobi operator J we obtain the estimate of the Lebesgue measure of the spectrum $$\mathrm{mes}(\sigma (J))\leqslant 4\min _{n}\mathop{ \mathrm{Tr}}\nolimits (a_{n}a_{n}^{{\ast}})^{\frac{1} {2} }$$, where a n are off-diagonal elements of J.
Anton A. Kutsenko

### Identifying Key Nodes in Social Networks Using Multi-Criteria Decision-Making Tools

This study investigates one of the major challenges in analysis of social networks: the identification of key nodes or important actors. There are numerous algorithms and approaches to locating and ranking nodes that may be critical in processes such as influence and diffusion. Most of the current algorithms consider a single criterion like the degree or page-rank of the nodes. However, many real world applications with large networks that display local sub-structure no single criterion may be adequate. We briefly discuss some single criteria that are often used to assess how node importance. Then a multiple-criteria decision-making method algorithm, TOPSIS, is presented. The proposed algorithm is examined on three datasets with varying size and sub-structure. Comparison of the results to those of other ranking algorithms such as PageRank indicate the ability of the suggested multi-criteria method to unambiguously rank nodes while remaining sensitive to the multiple ways in which a node may be “important”.

### On Band-Gap Structure of Spectrum in Network Double-Porosity Models

We study spectral properties of periodic divergent-type elliptic operator $$A_{\varepsilon }$$ with high contrast coefficients on $$\varepsilon$$-periodic thin network $$F_{\varepsilon }$$, which is asymptotically singular and can be obtained from 1-periodic graph F by means of fattening and contraction. The network $$F_{\varepsilon }$$ is divided into stiff and soft parts, also 1-periodic, where the coefficients of $$A_{\varepsilon }$$ are of order 1 and $$\varepsilon ^{2}$$, respectively; the stiff part is connected and the soft part is dispersive. We prove that the spectrum of the operator $$A_{\varepsilon }$$ has the band-gap structure and show the existence of non-degenerate spectral bands and open gaps, the number of which grows to infinity as $$\varepsilon \rightarrow 0$$. We establish connection between the endpoints of gaps and eigenvalues of two operators defined on the cell of periodicity. The first is the Laplace–Dirichlet operator on the “soft” part of the graph F within the unit cell and the second is its electrostatic extension onto the whole cell. Moreover, the band-gap structure of the spectrum can be described asymptotically exactly on each finite interval under additional geometric condition which is the “smallness” of the soft phase with respect to the stiff phase.
Svetlana E. Pastukhova

### Spectra, Energy and Laplacian Energy of Strong Double Graphs

For a graph G with vertex set $$V (G)\,=\,\{v_{1},v_{2},\cdots \,,v_{n}\}$$, the strong double graph SD(G) is a graph obtained by taking two copies of G and joining each vertex v i in one copy with the closed neighbourhood N[v i ] = N(v i ) ∪{ v i } of corresponding vertex in another copy. In this paper, we study spectra, energy and Laplacian energy of the graph SD(G). We also obtain some new families of equienergetic and L-equienergetic graphs, and an infinite family of graphs G for which LE(G) < E(G). We derive a formula for the number of spanning trees of SD(G) in terms of the number of spanning trees of G.

### System/Environment Duality of Nonequilibrium Network Observables

On networks representing probability currents between states of a system, we generalize Schnakenberg’s theory of nonequilibrium observables to nonsteady states, with the introduction of a new set of macroscopic observables that, for planar graphs, are related by a duality. We apply this duality to the linear regime, obtaining a dual proposition for the minimum entropy production principle, and to discrete electromagnetism, finding that it exchanges fields with sources. We interpret duality as reversing the role of system and environment, and discuss generalization to nonplanar graphs. The results are based on two theorems regarding the representation of bilinear and quadratic forms over the edge vector space of an oriented graph in terms of observables associated to cycles and cocycles.
Matteo Polettini
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