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fractional dynamics on networks

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fractional dynamics on networks

  1. 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. 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. 3. Description of networks
  4. 4. diffusion process and normal random walks Stochastic approach Deterministic approach structure! of network Adjacency ! matrix A Laplacian ! matrix L Transitian ! matrix W
  5. 5. how to make the dynamics fractional? non-fractional fractional Stochastic approach Deterministic approach
  6. 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. 7. Example 2: network on a ring (periodic) geodesic distance the Levy! measure of! a stable ! process !
  8. 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. 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?

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