Fractional Dynamics on Networks and Lattices - Thomas Michelitsch - ebook

Fractional Dynamics on Networks and Lattices ebook

Thomas Michelitsch

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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.

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Table of Contents

Cover

Preface

PART 1: Dynamics on General Networks

1 Characterization of Networks: the Laplacian Matrix and its Functions

1.1. Introduction

1.2. Graph theory and networks

1.3. Spectral properties of the Laplacian matrix

1.4. Functions that preserve the Laplacian structure

1.5. General properties of

g

(L)

1.6. Appendix: Laplacian eigenvalues for interacting cycles

2 The Fractional Laplacian of Networks

2.1. Introduction

2.2. General properties of the fractional Laplacian

2.3. Fractional Laplacian for regular graphs

2.4. Fractional Laplacian and type (i) and type (ii) functions

2.5. Appendix: Some basic properties of measures

3 Markovian Random Walks on Undirected Networks

3.1. Introduction

3.2. Ergodic Markov chains and random walks on graphs

3.3. Appendix: further spectral properties of the transition matrix Π

3.4. Appendix: Markov chains and bipartite networks

4 Random Walks with Long-range Steps on Networks

4.1. Introduction

4.2. Random walk strategies and

g

(L)

4.3. Lévy flights on networks

4.4. Transition matrix for types (i) and (ii) Laplacian functions

4.5. Global characterization of random walk strategies

4.6. Final remarks

4.7. Appendix: Functions

g

(L) for infinite one-dimensional lattices

4.8. Appendix: Positiveness of the generalized degree in regular networks

5 Fractional Classical and Quantum Transport on Networks

5.1. Introduction

5.2. Fractional classical transport on networks

5.3. Fractional quantum transport on networks

PART 2: Dynamics on Lattices

6 Explicit Evaluation of the Fractional Laplacian Matrix of Rings

6.1. Introduction

6.2. The fractional Laplacian matrix on rings

6.3. Riesz fractional derivative continuum limit kernels of the Fractional Laplacian matrix

6.4. Concluding remarks

6.5. Appendix: fractional Laplacian matrix of the ring

6.6. Appendix: estimates for the fractional degree in regular networks

7 Recurrence and Transience of the “Fractional Random Walk”

7.1. Introduction

7.2. General random walk characteristics

7.3. Universal features of the FRW

7.4. Recurrence theorem for the fractional random walk on

d

-dimensional infinite lattices

7.5. Emergence of Lévy flights and asymptotic scaling laws

7.6. Fractal scaling of the set of distinct nodes ever visited

7.7. Transient regime 0 <

α

< 1 of FRW on the infinite ring

7.8. Concluding remarks

7.9. Appendix: Recurrence and transience of FRW

8 Asymptotic Behavior of Markovian Random Walks Generated by Laplacian Matrix Functions

8.1. Introduction

8.2. Markovian walks generated by type (i) and type (ii) Laplacian matrix functions

8.3. Continuum limits – infinite network limits

8.4. Appendix

References

Index

End User License Agreement

List of Tables

Chapter 2

Table 2.1. Structure of the Fullerene-26. We present the values of (A

n

)

ii

with n...

Guide

Cover

Table of Contents

Begin Reading

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Series Editor

Noël Challamel

Fractional Dynamics on Networks and Lattices

Thomas Michelitsch

Alejandro Pérez Riascos

Bernard Collet

Andrzej Nowakowski

Franck Nicolleau

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com

© ISTE Ltd 2019

The rights of Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski and Franck Nicolleau to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2019930611

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-158-1

PART 1Dynamics on General Networks