Fractional dynamics on networks:!anomalous diffusion!

and!Levy flights

To present at the fractional club meeting!on 10/27/2014 by Summer Zheng

Presenting the paper 'Fractional dynamics on networks: Emergence of anomalous diffusion and Levy flights' by A.P. Riascos and Jose L. Mateos.

Table of content2. Form

alism

of dynam

ics

on network

s

diffusion process!+!

normal random walks

fractional diffusion!+!

long-range dynamics

1. General terminologies !

for networks

Example 2: network!

on a ring

Example 1: network!

on a tree

Fractional return!probability

Description of networks

diffusion process and normal random walks

Stochastic approach

Deterministic approach

structure!of network

Adjacency !matrix A

Laplacian !matrix L

Transitian !matrix W

how to make the dynamics fractional?

non-fractional fractional

Stochastic approach

Deterministic approach

Example 1: network on a tree

structure!of network

Adjacency !matrix A

Laplacian !matrix L

Transitian !matrix W

Fractional!Laplacian

Observation: long-range dynamics/ global dynamics in networks

Example 2: network on a ring (periodic)

geodesic distance

the Levy!measure of!

a stable !process !

Average fractional return probability

versus topology of the network

asymptotic power law

Definition

a ring

a tree

a scale-free network!of the Brabasi-Albert type

the return probability !depends on the shapes !

of the network!

Global time also depends on shapes of networks

efficiency to explore the network

Questions:!1. is the emergence of Levy flight shape-dependent?!2. 'tempered' fractional dynamics on networks?!3. how to introduce 'spectral methods' onto networks?!4. how do probability quantities depend on the structure of networks?