fractional dynamics on networks
TRANSCRIPT
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?