1. Random walks on complex networks
Zhongzhi ZHANG (章忠志)
School of Computer Science, Fudan University
Homepage:

2. Contents Introduction to random walks on graphs
Applications of random walks
Trapping on graphs: Introduction and Review
Trapping on complex networks
Future work
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3. Random Walks on Graphs Kinds of random walks:-
Kinds of random walks:
Unbiased random walks, biased random walks, self-avoid walks, quantum walks, ect
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4. Random Walks on Graphs -
At any node, go to one of the neighbors of the node with equal probability.
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5. Random Walks on Graphs -
At any node, go to one of the neighbors of the node with equal probability.
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6. Random Walks on Graphs -
At any node, go to one of the neighbors of the node with equal probability.
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7. Random Walks on Graphs -
At any node, go to one of the neighbors of the node with equal probability.
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8. Random Walks on Graphs -
At any node, go to one of the neighbors of the node with equal probability.
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9. Important parameters of random walks
Primary
Indicators
First-Passage Time F(s,t): Expected number of steps to reach t starting at s
Mean Commute time C(s,t): Steps from s to t, and then go back C(t,s) = F(s,t) + F(t,s)
Mean Return time T(s,s): mean time for returning to node s for the first time after having left it
Cover time, survival problity, ……
New J. Phys. 7, 26 (2005)
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10. FPT for general graphs Pseudoinverse of the Laplacian matrix
N: Node number
is the Laplacian matrix defined as
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11. FPT for complete graph
Study random walks on simple structure is of exceptional importance.
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12. FPT for a linear graph
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13. Applications of random walks
Community detection
Recommendation systems
Electrical networks
Spanning trees
Information Retrieval
Natural Language Processing
Machine Learning
Graph partitioning
In economics: random walk hypothesis
PageRank algorithm
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14. Application to Community detection
World Wide Web
Protein interaction networks
Metabolic networks
Social networks
Biological networks
Food Webs
Properties of community may be quite different from the average property of network.
More links “inside” than “outside”
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15. Application to recommendation systems
Help for setting up efficient recommendation systems
IEEE Trans. Knowl. Data Eng. 19, 355 (2007)
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16. Connections with electrical networks
Every edge – a resistor of 1 ohm.
Voltage difference of 1 volt between u and v.
R(u,v) – inverse of electrical current from u to v. v _ + u
C(u,v) = F(s,t) + F(t,s) =2mR(u,v), R(u,v)= C(u,v)/ (2m)
dz is degree of z, m is the number of edges
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17. Formulas for effective resistance
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18. Random walks and spanning trees
EPL (Europhysics Letters), 2010, 90:68002
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19. Trapping problem: Random walks on graphs with an immobile trap
Trapping time for node i denoted by
Average trapping time
Research goal: obtain the dependence of average trapping time on the system size
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20. Previous work Trapping on Sierpinski gasket
Physical Review E, 2002, 65:
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21. Previous work: Trapping on T-fractal
Physical Review E, 2008, 77:
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22. Previous work: Trapping on regular lattices
Journal of Mathematical Physics, 1969, 10: 753
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23. Our work Random walks on scale-free networks
A pseudofractal scale-free web
Apollonian networks
Modular scale-free networks
Koch networks
A fractal scale-free network
Scale-free networks with the same degree sequences
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24. Main contributions Unveil the effect of trap location on ATT
Methods for finding Mean first-passage time (MFPT)
Backward equations Generating functions
Laplacian spectra Electrical networks
Uncover the impacts of structures on MFPT
Scale-free behavior
Modular structure
Fractal structure
Unveil the effect of trap location on ATT
Relate random walks to Hanoi Towers Game
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25. Walks on pseudofractal scale-free web
Physical Review E, 2009, 79:
Contributions：(1)New method (2)New findings
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26. Walks on Apollonian network
The most efficient structure for diffusion
EPL, 2009, 86:
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27. Walks on modular scale-free networks
Generating function method
Physical Review E, 2009, 80:
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28. ATT on uncorrelated scale-free networks
In uncorrelated random SF webs,
Physical Review E, 2009, 79:
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29. Walks on the Koch network
Physical Review E, 2009, 79:
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30. Impact of degree heterogeneity on MFPT
Journal of Physics A, 2010, 43:
Chaos, 2010, 20:
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31. Effect of fractality on MFPT
Definition of fractal networks
Protein interaction network
Box-covering method
WWW
Metabolic network
Mean mass (number of nodes) within a box:
Hollywood film actors network
Last year Song et al. discovered the fractal scaling in scale-free networks. They found that the number of boxes to cover a given network scales in a power law with the box size ell. The mean number of nodes within a box scales in a power law. This power-law behavior seems to be contradictory to the small-world feature. The number of nodes with a distance ell from a given vertex increases exponentially with ell. The box size is defined as the maximum distance within a box.
Song, Havlin, and Makse, Nature (2005).
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32. Walks on a fractal scale-free network
EPL (Europhysics Letters), 2009, 88:
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33. Walks on scale-free networks with identical degree sequences
Physical Review E, 2009, 79:
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34. Walks on scale-free networks with identical degree sequences
Advantages： (1) without crossing edges (2) always connected.
Physical Review E, 2009, 80:
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35. Influence of trap position on MFPT on various networks
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36. Significant impact of trap position on MFPT in non-fractal scale-free trees
Journal of Mathematical Physics, 2009, 50:
Journal of Physics A, 2011, 44:
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37. Global mean first-passage time
Significant impact of trap position on MFPT in scale-free Koch networks
Mean receiving time
Mean sending time
Global mean first-passage time
European Physical Journal B, 2011, (in press).
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38. No essential impact of trap position on MFPT in fractal scale-free trees
Journal of Physics A, 2011, 44:
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39. No qualitative effect of trap location on MFPT in extended T-graphs
Physical Review E, 2008, 77:
Physical Review E, 2010, 82:
New Journal of Physics, 2009, 11:
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40. Random Walks on Vicsek fractals
Physical Review E, 2010, 81:
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41. Random Walks on dual Sierpinski gasket
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42. Relation to the Hanoi Towers Game
What is the minimum number of moves ?
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43. The Hanoi Towers Graphs
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44. Future work Walks with multiple traps 1 Quantum walks on networks 2
Self-avoid walks 3
Biased walks, e.g. walks on weighted nets 4
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45. Thank You!