Online resource; title from PDF title page (EBSCO, viewed April 15, 2019).

Includes bibliographical references and index.

Cover; Half-Title Page; Title Page; Copyright Page; Contents; 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; 1.2.1. Basic graph theory; 1.2.2. Networks; 1.3. Spectral properties of the Laplacian matrix; 1.3.1. Laplacian matrix; 1.3.2. General properties of the Laplacian eigenvalues and eigenvectors; 1.3.3. Spectra of some typical graphs; 1.4. Functions that preserve the Laplacian structure; 1.4.1. Function g(L) and general conditions

1.4.2. Non-negative symmetric matrices1.4.3. Completely monotonic functions; 1.5. General properties of g(L); 1.5.1. Diagonal elements (generalized degree); 1.5.2. Functions g(L) for regular graphs; 1.5.3. Locality and non-locality of g(L) in the limit of large networks; 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 measures3. Markovian Random Walks on Undirected Networks; 3.1. Introduction; 3.2. Ergodic Markov chains and random walks on graphs; 3.2.1. Characterization of networks: the Laplacian matrix; 3.2.2. Characterization of random walks on networks: Ergodic Markov chains; 3.2.3. The fundamental theorem of Markov chains; 3.2.4. The ergodic hypothesis and theorem; 3.2.5. Strong law of large numbers; 3.2.6. Analysis of the spectral properties of the transition matrix; 3.3. Appendix: further spectral properties of the transition matrix

3.4. Appendix: Markov chains and bipartite networks3.4.1. Unique overall probability in bipartite networks; 3.4.2. Eigenvalue structure of the transition matrix for normal walks in bipartite graphs; 4. Random Walks with Long-range Steps on Networks; 4.1. Introduction; 4.2. Random walk strategies and; 4.2.1. Fractional Laplacian; 4.2.2. Logarithmic functions of the Laplacian; 4.2.3. Exponential functions of the Laplacian; 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

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 LEvy flights. In Part 2, fractional dynamics and LEvy flight behavior are analyzed thoroughly, and a generalization of POlya'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.