Fractional Dynamics on Networks and Lattices

Thomas Michelitsch, Alejandro Perez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau

• Publisher: Wiley-ISTE
• Year: 2019
• Print ISBN: 9781786301581
• eBook ISBN: 9781119608219
• Edition: 1
• Topic: Technik & Maschinenbau
• Subtopic: Technik, Forschung & Kompetenzen

This book analyzes stochastic processes on networks and regular structures such as lattices by employing the Markovian random walk approach.

Part 1 is devoted to the study of local and non-local random walks. It shows how non-local random walk strategies can be defined by functions of the Laplacian matrix that maintain the stochasticity of the transition probabilities. A major result is that only two types of functions are admissible: type (i) functions generate asymptotically local walks with the emergence of Brownian motion, whereas type (ii) functions generate asymptotically scale-free non-local "fractional" walks with the emergence of Lévy flights.

In Part 2, fractional dynamics and Lévy flight behavior are analyzed thoroughly, and a generalization of Pólya's classical recurrence theorem is developed for fractional walks. The authors analyze primary fractional walk characteristics such as the mean occupation time, the mean first passage time, the fractal scaling of the set of distinct nodes visited, etc. The results show the improved search capacities of fractional dynamics on networks.

PART 1
Dynamics on General Networks

1
Characterization of Networks: the Laplacian Matrix and its Functions

1.1. Introduction

The study of networks, their characteristics and dynamical processes taking place on these structures have had a significant impact in different fields of science and engineering, leading to important applications in the context of physics, biology and social and computer systems among many others. In this chapter, we present an introduction to several definitions in the context of the study of undirected connected networks that are used and discussed in various parts of this book. We start with an introduction to graph theory and concepts related to the connectivity of networks, in particular, the concept of distance in networks and the average of this quantity that gives a global characterization of the network connectivity. Different types of networks and their characteristics are described as well as three common algorithms to generate random networks.

In the second part of this chapter, the Laplacian matrix L of a network is discussed along with general properties of the eigenvalues and the respective eigenvectors of this matrix. The Laplacian matrix of a network has been explored in connection with dynamical processes on networks, in particular, diffusive transport and synchronization. Then we introduce a generalization of the notion of the “Laplacian matrix” and study a class of matrix functions g(L) of the Laplacian matrix that maintains its structure and general “good” properties. We demonstrate that this generalization allows describing a rich variety of new dynamic processes that cannot be captured by the Laplacian matrix. In the framework of this generalization, we introduce the concept of the fractional Laplacian matrix, which is explored in detail in Chapter 2, and we work in terms of general Laplacian matrix functions in Chapter 4. In this way, we will define several types of random walk strategies with long-range displacements on networks.

1.2. Graph theory and networks

1.2.1. Basic graph theory

In order to study dynamical processes taking pla...

Table of contents

• Cover
• Table of Contents
• Preface
• PART 1: Dynamics on General Networks
• PART 2: Dynamics on Lattices
• References
• Index
• End User License Agreement