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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

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
Creator
Contributor
Subject
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 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
Member of
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
Instantiates
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references and index
Color
multicolored
Contents
  • 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
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
Publication
Antecedent source
unknown
Bibliography note
Includes bibliographical references and index
Color
multicolored
Contents
  • 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
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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