for fractional club meeting at brown on 10/28/2014

1. 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.

2. Table of content
dynamics
Formalism networks
of on 2. diffusion process!
+!
normal random walks
fractional diffusion!
+!
long-range dynamics
1. General for networks
terminologies !
Example on a 1: tree
network!
Example on a 2: ring
network!
Fractional return!
probability

3. Description of networks

4. diffusion process and normal random walks
Stochastic approach
Deterministic approach
structure!
of network
Adjacency !
matrix A
Laplacian !
matrix L
Transitian !
matrix W

5. how to make the dynamics fractional?
non-fractional fractional
Stochastic approach
Deterministic approach

6. 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

7. Example 2: network on a ring (periodic)
geodesic distance
the Levy!
measure of!
a stable !
process !

8. Average fractional return probability
versus topology of the network
Definition
asymptotic power law
a ring
a tree
a scale-free network!
of the Brabasi-Albert type
the return probability !
depends on the shapes !
of the network!

9. 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?