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.

• Preface ixPart 1 Dynamics on General Networks 1Chapter 1 Characterization of Networks: the Laplacian Matrix and its Functions 31.1. Introduction 31.2. Graph theory and networks 41.2.1. Basic graph theory 41.2.2. Networks 61.3. Spectral properties of the Laplacian matrix 111.3.1. Laplacian matrix 111.3.2. General properties of the Laplacian eigenvalues and eigenvectors 131.3.3. Spectra of some typical graphs 151.4. Functions that preserve the Laplacian structure 171.4.1. Function g(L) and general conditions 171.4.2. Non-negative symmetric matrices 201.4.3. Completely monotonic functions 221.5 General properties of g(L) 281.5.1. Diagonal elements (generalized degree) 291.5.2. Functions g(L) for regular graphs 291.5.3. Locality and non-locality of g(L) in the limit of large networks 301.6. Appendix: Laplacian eigenvalues for interacting cycles 32Chapter 2 The Fractional Laplacian of Networks 332.1. Introduction 332.2. General properties of the fractional Laplacian 342.3. Fractional Laplacian for regular graphs 362.4. Fractional Laplacian and type (i) and type (ii) functions 412.5. Appendix: Some basic properties of measures 48Chapter 3 Markovian Random Walks on Undirected Networks 553.1. Introduction 553.2. Ergodic Markov chains and random walks on graphs 573.2.1. Characterization of networks: the Laplacian matrix 573.2.2. Characterization of random walks on networks: Ergodic Markov chains 583.2.3. The fundamental theorem of Markov chains 633.2.4. The ergodic hypothesis and theorem 683.2.5. Strong law of large numbers 753.2.6. Analysis of the spectral properties of the transition matrix 773.3 Appendix: further spectral properties of the transition matrix Π 823.4. Appendix: Markov chains and bipartite networks 843.4.1. Unique overall probability in bipartite networks 843.4.2. Eigenvalue structure of the transition matrix for normal walks in bipartite graphs 85Chapter 4 Random Walks with Long-range Steps on Networks 934.1. Introduction 934.2 Random walk strategies and g(L) 944.2.1. Fractional Laplacian 954.2.2. Logarithmic functions of the Laplacian 974.2.3. Exponential functions of the Laplacian 984.3. Lévy flights on networks 994.4. Transition matrix for types (i) and (ii) Laplacian functions 1024.5. Global characterization of random walk strategies 1054.5.1. Kemeny’s constant for finite rings 1084.5.2. Global time τ for irregular networks 1104.6. Final remarks 1124.7. Appendix: Functions g(L) for infinite one-dimensional lattices 1134.8. Appendix: Positiveness of the generalized degree in regular networks 114Chapter 5 Fractional Classical and Quantum Transport on Networks 1175.1. Introduction 1175.2. Fractional classical transport on networks 1185.2.1. Fractional diffusion equation 1185.2.2. Diffusion equation and random walks on networks 1205.2.3. Fractional random walks with continuous time 1225.2.4. Fractional average probability of return in an infinite ring 1255.2.5 Probability pn(γ)(t) for a ring in the limit N → ∞ 1275.2.6. Efficiency of the fractional diffusive transport 1295.3. Fractional quantum transport on networks 1335.3.1. Continuous-time quantum walks 1345.3.2. Fractional Schrödinger equation 1355.3.3. Fractional quantum walks 1355.3.4. Fractional quantum dynamics on interacting cycles 1365.3.5. Quantum transport on an infinite ring 1385.3.6. Efficiency of the fractional quantum transport 141Part 2 Dynamics on Lattices 143Chapter 6 Explicit Evaluation of the Fractional Laplacian Matrix of Rings 1456.1. Introduction 1456.2. The fractional Laplacian matrix on rings 1466.2.1. Preliminaries 1466.2.2. Explicit evaluation of the fractional Laplacian matrix for the infinite ring 1496.2.3. Fractional Laplacian of the finite ring 1546.3. Riesz fractional derivative continuum limit kernels of the Fractional Laplacian matrix 1556.3.1. General continuum limit procedure 1566.3.2. Infinite space continuum limit 1616.3.3. Periodic string continuum limit 1636.4. Concluding remarks 1656.5. Appendix: fractional Laplacian matrix of the ring 1666.5.1. Euler’s reflection formula 1706.5.2. Some useful relations for the infinite ring limit 1716.5.3. Asymptotic behavior of the fractional Laplacian matrix 1746.5.4. Canonic representations of the fractional Laplacian in the periodic string (i) and infinite space limit (ii) 1776.6. Appendix: estimates for the fractional degree in regular networks 179Chapter 7 Recurrence and Transience of the “Fractional Random Walk” 1837.1. Introduction 1837.2. General random walk characteristics 1877.2.1. Mean occupation times, long-range moves and first passage quantities 1877.2.2. Probability generating functions and recurrence behavior 1967.3. Universal features of the FRW 2037.4. Recurrence theorem for the fractional random walk on d-dimensional infinite lattices 2087.5. Emergence of Lévy flights and asymptotic scaling laws 2167.6. Fractal scaling of the set of distinct nodes ever visited 2207.7. Transient regime 0

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.

• Preface ixPart 1 Dynamics on General Networks 1Chapter 1 Characterization of Networks: the Laplacian Matrix and its Functions 31.1. Introduction 31.2. Graph theory and networks 41.2.1. Basic graph theory 41.2.2. Networks 61.3. Spectral properties of the Laplacian matrix 111.3.1. Laplacian matrix 111.3.2. General properties of the Laplacian eigenvalues and eigenvectors 131.3.3. Spectra of some typical graphs 151.4. Functions that preserve the Laplacian structure 171.4.1. Function g(L) and general conditions 171.4.2. Non-negative symmetric matrices 201.4.3. Completely monotonic functions 221.5 General properties of g(L) 281.5.1. Diagonal elements (generalized degree) 291.5.2. Functions g(L) for regular graphs 291.5.3. Locality and non-locality of g(L) in the limit of large networks 301.6. Appendix: Laplacian eigenvalues for interacting cycles 32Chapter 2 The Fractional Laplacian of Networks 332.1. Introduction 332.2. General properties of the fractional Laplacian 342.3. Fractional Laplacian for regular graphs 362.4. Fractional Laplacian and type (i) and type (ii) functions 412.5. Appendix: Some basic properties of measures 48Chapter 3 Markovian Random Walks on Undirected Networks 553.1. Introduction 553.2. Ergodic Markov chains and random walks on graphs 573.2.1. Characterization of networks: the Laplacian matrix 573.2.2. Characterization of random walks on networks: Ergodic Markov chains 583.2.3. The fundamental theorem of Markov chains 633.2.4. The ergodic hypothesis and theorem 683.2.5. Strong law of large numbers 753.2.6. Analysis of the spectral properties of the transition matrix 773.3 Appendix: further spectral properties of the transition matrix Π 823.4. Appendix: Markov chains and bipartite networks 843.4.1. Unique overall probability in bipartite networks 843.4.2. Eigenvalue structure of the transition matrix for normal walks in bipartite graphs 85Chapter 4 Random Walks with Long-range Steps on Networks 934.1. Introduction 934.2 Random walk strategies and g(L) 944.2.1. Fractional Laplacian 954.2.2. Logarithmic functions of the Laplacian 974.2.3. Exponential functions of the Laplacian 984.3. Lévy flights on networks 994.4. Transition matrix for types (i) and (ii) Laplacian functions 1024.5. Global characterization of random walk strategies 1054.5.1. Kemeny’s constant for finite rings 1084.5.2. Global time τ for irregular networks 1104.6. Final remarks 1124.7. Appendix: Functions g(L) for infinite one-dimensional lattices 1134.8. Appendix: Positiveness of the generalized degree in regular networks 114Chapter 5 Fractional Classical and Quantum Transport on Networks 1175.1. Introduction 1175.2. Fractional classical transport on networks 1185.2.1. Fractional diffusion equation 1185.2.2. Diffusion equation and random walks on networks 1205.2.3. Fractional random walks with continuous time 1225.2.4. Fractional average probability of return in an infinite ring 1255.2.5 Probability pn(γ)(t) for a ring in the limit N → ∞ 1275.2.6. Efficiency of the fractional diffusive transport 1295.3. Fractional quantum transport on networks 1335.3.1. Continuous-time quantum walks 1345.3.2. Fractional Schrödinger equation 1355.3.3. Fractional quantum walks 1355.3.4. Fractional quantum dynamics on interacting cycles 1365.3.5. Quantum transport on an infinite ring 1385.3.6. Efficiency of the fractional quantum transport 141Part 2 Dynamics on Lattices 143Chapter 6 Explicit Evaluation of the Fractional Laplacian Matrix of Rings 1456.1. Introduction 1456.2. The fractional Laplacian matrix on rings 1466.2.1. Preliminaries 1466.2.2. Explicit evaluation of the fractional Laplacian matrix for the infinite ring 1496.2.3. Fractional Laplacian of the finite ring 1546.3. Riesz fractional derivative continuum limit kernels of the Fractional Laplacian matrix 1556.3.1. General continuum limit procedure 1566.3.2. Infinite space continuum limit 1616.3.3. Periodic string continuum limit 1636.4. Concluding remarks 1656.5. Appendix: fractional Laplacian matrix of the ring 1666.5.1. Euler’s reflection formula 1706.5.2. Some useful relations for the infinite ring limit 1716.5.3. Asymptotic behavior of the fractional Laplacian matrix 1746.5.4. Canonic representations of the fractional Laplacian in the periodic string (i) and infinite space limit (ii) 1776.6. Appendix: estimates for the fractional degree in regular networks 179Chapter 7 Recurrence and Transience of the “Fractional Random Walk” 1837.1. Introduction 1837.2. General random walk characteristics 1877.2.1. Mean occupation times, long-range moves and first passage quantities 1877.2.2. Probability generating functions and recurrence behavior 1967.3. Universal features of the FRW 2037.4. Recurrence theorem for the fractional random walk on d-dimensional infinite lattices 2087.5. Emergence of Lévy flights and asymptotic scaling laws 2167.6. Fractal scaling of the set of distinct nodes ever visited 2207.7. Transient regime 0