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E-grāmata: Fractional Dynamics on Lattices and Networks [Wiley Online]

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  • Formāts: 336 pages
  • Izdošanas datums: 12-Apr-2019
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • ISBN-10: 1119608163
  • ISBN-13: 9781119608165
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Fractional Dynamics on Lattices and NetworksPalielināt
  • Formāts: 336 pages
  • Izdošanas datums: 12-Apr-2019
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • ISBN-10: 1119608163
  • ISBN-13: 9781119608165
Citas grāmatas par šo tēmu:
Wiley Online E-booksMore info about Wiley Online
Rokasgrāmatas mājas lapa: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119608165

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.

Preface ix
Part 1 Dynamics on General Networks
1(142)
Chapter 1 Characterization of Networks: the Laplacian Matrix and its Functions
3(30)
1.1 Introduction
3(1)
1.2 Graph theory and networks
4(7)
1.2.1 Basic graph theory
4(2)
1.2.2 Networks
6(5)
1.3 Spectral properties of the Laplacian matrix
11(6)
1.3.1 Laplacian matrix
11(2)
1.3.2 General properties of the Laplacian eigenvalues and eigenvectors
13(2)
1.3.3 Spectra of some typical graphs
15(2)
1.4 Functions that preserve the Laplacian structure
17(11)
1.4.1 Function g(L) and general conditions
17(3)
1.4.2 Non-negative symmetric matrices
20(2)
1.4.3 Completely monotonic functions
22(6)
1.5 General properties of g(L)
28(4)
1.5.1 Diagonal elements (generalized degree)
29(1)
1.5.2 Functions g(L) for regular graphs
29(1)
1.5.3 Locality and non-locality of g(L) in the limit of large networks
30(2)
1.6 Appendix: Laplacian eigenvalues for interacting cycles
32(1)
Chapter 2 The Fractional Laplacian of Networks
33(22)
2.1 Introduction
33(1)
2.2 General properties of the fractional Laplacian
34(2)
2.3 Fractional Laplacian for regular graphs
36(5)
2.4 Fractional Laplacian and type (i) and type (ii) functions
41(7)
2.5 Appendix: Some basic properties of measures
48(7)
Chapter 3 Markovian Random Walks on Undirected Networks
55(38)
3.1 Introduction
55(2)
3.2 Ergodic Markov chains and random walks on graphs
57(25)
3.2.1 Characterization of networks: the Laplacian matrix
57(1)
3.2.2 Characterization of random walks on networks: Ergodic Markov chains
58(5)
3.2.3 The fundamental theorem of Markov chains
63(5)
3.2.4 The ergodic hypothesis and theorem
68(7)
3.2.5 Strong law of large numbers
75(2)
3.2.6 Analysis of the spectral properties of the transition matrix
77(5)
3.3 Appendix: further spectral properties of the transition matrix Π
82(2)
3.4 Appendix: Markov chains and bipartite networks
84(9)
3.4.1 Unique overall probability in bipartite networks
84(1)
3.4.2 Eigenvalue structure of the transition matrix for normal walks in bipartite graphs
85(8)
Chapter 4 Random Walks with Long-range Steps on Networks
93(24)
4.1 Introduction
93(1)
4.2 Random walk strategies and g(L)
94(5)
4.2.1 Fractional Laplacian
95(2)
4.2.2 Logarithmic functions of the Laplacian
97(1)
4.2.3 Exponential functions of the Laplacian
98(1)
4.3 Levy nights on networks
99(3)
4.4 Transition matrix for types (i) and (ii) Laplacian functions
102(3)
4.5 Global characterization of random walk strategies
105(7)
4.5.1 Kemeny's constant for finite rings
108(2)
4.5.2 Global time τ for irregular networks
110(2)
4.6 Final remarks
112(1)
4.7 Appendix: Functions g(L) for infinite one-dimensional lattices
113(1)
4.8 Appendix: Positiveness of the generalized degree in regular networks
114(3)
Chapter 5 Fractional Classical and Quantum Transport on Networks
117(26)
5.1 Introduction
117(1)
5.2 Fractional classical transport on networks
118(15)
5.2.1 Fractional diffusion equation
118(2)
5.2.2 Diffusion equation and random walks on networks
120(2)
5.2.3 Fractional random walks with continuous time
122(3)
5.2.4 Fractional average probability of return in an infinite ring
125(2)
5.2.5 Probability p(γ)n(t) for a ring in the limit N → ∞
127(2)
5.2.6 Efficiency of the fractional diffusive transport
129(4)
5.3 Fractional quantum transport on networks
133(10)
5.3.1 Continuous-time quantum walks
134(1)
5.3.2 Fractional Schrodinger equation
135(1)
5.3.3 Fractional quantum walks
135(1)
5.3.4 Fractional quantum dynamics on interacting cycles
136(2)
5.3.5 Quantum transport on an infinite ring
138(3)
5.3.6 Efficiency of the fractional quantum transport
141(2)
Part 2 Dynamics on Lattices
143(150)
Chapter 6 Explicit Evaluation of the Fractional Laplacian Matrix of Rings
145(38)
6.1 Introduction
145(1)
6.2 The fractional Laplacian matrix on rings
146(9)
6.2.1 Preliminaries
146(3)
6.2.2 Explicit evaluation of the fractional Laplacian matrix for the infinite ring
149(5)
6.2.3 Fractional Laplacian of the finite ring
154(1)
6.3 Riesz fractional derivative continuum limit kernels of the Fractional Laplacian matrix
155(10)
6.3.1 General continuum limit procedure
156(5)
6.3.2 Infinite space continuum limit
161(2)
6.3.3 Periodic string continuum limit
163(2)
6.4 Concluding remarks
165(1)
6.5 Appendix: fractional Laplacian matrix of the ring
166(13)
6.5.1 Euler's reflection formula
170(1)
6.5.2 Some useful relations for the infinite ring limit
171(3)
6.5.3 Asymptotic behavior of the fractional Laplacian matrix
174(3)
6.5.4 Canonic representations of the fractional Laplacian in the periodic string (i) and infinite space limit (ii)
177(2)
6.6 Appendix: estimates for the fractional degree in regular networks
179(4)
Chapter 7 Recurrence and Transience of the "Fractional Random Walk"
183(56)
7.1 Introduction
183(4)
7.2 General random walk characteristics
187(16)
7.2.1 Mean occupation times, long-range moves and first passage quantities
187(9)
7.2.2 Probability generating functions and recurrence behavior
196(7)
7.3 Universal features of the FRW
203(5)
7.4 Recurrence theorem for the fractional random walk on d-dimensional infinite lattices
208(8)
7.5 Emergence of Levy flights and asymptotic scaling laws
216(4)
7.6 Fractal scaling of the set of distinct nodes ever visited
220(6)
7.7 Transient regime 0 < α < 1 of FRW on the infinite ring
226(7)
7.8 Concluding remarks
233(2)
7.9 Appendix: Recurrence and transience of FRW
235(4)
7.9.1 Properties of F(α)|p|
235(1)
7.9.2 Recurrent limits
236(3)
Chapter 8 Asymptotic Behavior of Markovian Random Walks Generated by Laplacian Matrix Functions
239(54)
8.1 Introduction
239(4)
8.2 Markovian walks generated by type (i) and type (ii) Laplacian matrix functions
243(3)
8.3 Continuum limits -- infinite network limits
246(27)
8.3.1 The Pearson walk
251(4)
8.3.2 Type (i) Laplacian kernels: Emergence of Brownian motion (Rayleigh flights) and normal diffusion
255(5)
8.3.3 Type (ii) Laplacian density kernels: Emergence of Levy flights and anomalous diffusion
260(6)
8.3.4 Green's function -- MRT
266(4)
8.3.5 Some brief remarks on self-similar fractal distributions of nodes
270(3)
8.4 Appendix
273(20)
8.4.1 Emergence of symmetric α-stable limiting transition PDFs
273(4)
8.4.2 Some properties of symmetric α-stable PDFs
277(5)
8.4.3 Spectral dimension of the FRW -- Levy flight
282(2)
8.4.4 Evaluation of some integrals and normalization constants of the fractional Laplacian
284(5)
8.4.5 Regularization and further properties of the fractional Laplacian kernel
289(4)
References 293(10)
Index 303
Thomas Michelitsch is a CNRS Senior Research Scientist at the Institut Jean le Rond d'Alembert, Sorbonne University, France.

Alejandro Pérez Riascos is a Researcher and Associate Professor at the Institute of Physics at the Universidad Nacional Autonoma de México.

Bernard Collet is Emeritus Professor at the Institut Jean le Rond d'Alembert, Sorbonne University, France.

Andrzej Nowakowski is a Lecturer at the University of Sheffield, UK.

Franck Nicolleau is a Senior Lecturer at the University of Sheffield, UK, and Head of the Sheffield Fluid Mechanics Group.
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