Fractional Dynamics on Networks and Lattices

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.

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.

Part 1. Dynamics on General Networks

 1. Characterization of Networks: the Laplacian Matrix and its Functions.

 2. The Fractional Laplacian of Networks.

 3. Markovian Random Walks on Undirected Networks.

 4. Random Walks with Long-range Steps on Networks.

 5. Fractional Classical and Quantum Transport on Networks.

 Part 2. Dynamics on Lattices

 6. Explicit Evaluation of the Fractional Laplacian Matrix of Rings.

 7. Recurrence and Transience of the “Fractional Random Walk”.

8. Asymptotic Behavior of Markovian Random Walks Generated by Laplacian Matrix Functions.

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