The Resource Fractional dynamics on networks and lattices, Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau
Fractional dynamics on networks and lattices, Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau
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The item Fractional dynamics on networks and lattices, Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Massey University Library, University of New Zealand.This item is available to borrow from 1 library branch.
Resource Information
The item Fractional dynamics on networks and lattices, Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Massey University Library, University of New Zealand.
This item is available to borrow from 1 library branch.
 Summary
 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 nonlocal random walks. It shows how nonlocal 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 scalefree nonlocal "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
 Language
 eng
 Extent
 1 online resource
 Contents

 Cover; HalfTitle 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. Nonnegative 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 nonlocality 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 Longrange 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
 Isbn
 9781119608165
 Label
 Fractional dynamics on networks and lattices
 Title
 Fractional dynamics on networks and lattices
 Statement of responsibility
 Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau
 Language
 eng
 Summary
 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 nonlocal random walks. It shows how nonlocal 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 scalefree nonlocal "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
 http://library.link/vocab/creatorName
 Michelitsch, Thomas,
 Dewey number
 519.2/33
 Illustrations
 illustrations
 Index
 index present
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 http://library.link/vocab/relatedWorkOrContributorName

 Pérez Riascos, Alejandro,
 Collet, Bernard,
 Nowakowski, Andrzej,
 Nicolleau, F. C. G. A.
 Series statement
 Mechanical engineering and solid mechanics series
 http://library.link/vocab/subjectName

 Markov processes
 Random walks (Mathematics)
 Label
 Fractional dynamics on networks and lattices, Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Color
 multicolored
 Contents

 Cover; HalfTitle 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. Nonnegative 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 nonlocality 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 Longrange 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
 Control code
 on1096435591
 Dimensions
 unknown
 Extent
 1 online resource
 File format
 unknown
 Form of item
 online
 Isbn
 9781119608165
 Level of compression
 unknown
 Note
 John Wiley and Sons
 Other physical details
 illustrations
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)1096435591
 Label
 Fractional dynamics on networks and lattices, Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau
 Antecedent source
 unknown
 Bibliography note
 Includes bibliographical references and index
 Color
 multicolored
 Contents

 Cover; HalfTitle 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. Nonnegative 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 nonlocality 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 Longrange 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
 Control code
 on1096435591
 Dimensions
 unknown
 Extent
 1 online resource
 File format
 unknown
 Form of item
 online
 Isbn
 9781119608165
 Level of compression
 unknown
 Note
 John Wiley and Sons
 Other physical details
 illustrations
 Quality assurance targets
 not applicable
 Reformatting quality
 unknown
 Sound
 unknown sound
 Specific material designation
 remote
 System control number
 (OCoLC)1096435591
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