1. 报告人： 林 苑 指导老师：章忠志 副教授 复旦大学 2010.10.17
第六届全国复杂网络会议 CCCN2010
Trapping in scale-free networks with hierarchical organization of modularity
报告人： 林 苑
指导老师：章忠志 副教授
复旦大学
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2. Outline Introduction about random walks Our works Concepts
Applications
Our works
Fixed-trap problem
Multi-trap problem
Hamiltonian walks
Self-avoid walks
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3. Random walks
At any node, go to one of the neighbors of the node with equal probability. -
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4. Random walks
At any node, go to one of the neighbors of the node with equal probability. -
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5. Random walks
At any node, go to one of the neighbors of the node with equal probability. -
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6. Random walks
At any node, go to one of the neighbors of the node with equal probability. -
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7. Random walks
At any node, go to one of the neighbors of the node with equal probability. -
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8. Random walks Random walks can be depicted accurately by Markov Chain.
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9. Generic Approach Markov Chain Generating Function Laplacian matrix
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10. Measures Mean transit time Tij Mean return time Tii
Tij ≠ Tji
Mean return time Tii
Mean commute time Cij
Cij =Tij+Tji
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11. Applications PageRank of Google Cited time Semantic categorization
Recommendatory System
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12. Applications One major issue: How closed are two nodes?
Distance between nodes
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13. Applications Classical methods Based on Random Walk (or diffusion)
Shortest Path Length
Numbers of Paths
Based on Random Walk (or diffusion)
Mean transit time,
Mean commute time
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14. Applications The latter methods should be better, however…
Calculate inverse of matrix for O(|V|) times.
Need more efficient way to calculate.
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15. Trapping Problem
Imagine there are traps (or absorbers) on several certain vertices.
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16. Trapping Problem
Imagine there are traps (or absorbers) on several certain vertices.
We are interested the time of absorption.
For simplicity, we first consider the problem that only a single trap.
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17. Zhang Zhongzhi, Lin Yuan, et al. Physical Review E, 2009, 80: 051120.
Trapping in scale-free networks with hierarchical organization of modularity,
Zhang Zhongzhi, Lin Yuan, et al.
Physical Review E, 2009, 80:
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18. Two remarkable features
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19. Two remarkable features
Scale-free topology
Modular organization
For a large number of real networks, these two features coexist:
Protein interaction network
Metabolic networks
The World Wide Web
Some social networks
… …
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20. Trapping issue
Understand how the dynamical processes are influenced by the underlying topological structure.
Trapping issue relevant to a variety of contexts.
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21. Modular scale-free networks
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22. Modular scale-free networks
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23. Modular scale-free networks
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24. Modular scale-free networks
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25. Modular scale-free networks
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26. Modular scale-free networks
We denote by Hg the network model after g iterations.
For g=1,
The network consists of a central node, called the hub node,
And M-1 peripheral (external) nodes. All these M nodes are fully connected to each other.
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27. Modular scale-free networks
We denote by Hg the network model after g iterations.
For g>1,
Hg can be obtained from Hg-1 by adding M-1 replicas of Hg-1 with their external nodes being linked to the hub of original Hg-1 unit.
The new hub is the hub of original Hg-1 unit.
The new external nodes are composed of all the peripheral nodes of M-1 copies of Hg-1.
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28. Formulation of the trapping problemXi
First-passage time (FPT)
Markov chain
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29. Formulation of the trapping problem
Define a generating function
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30. Formulation of the trapping problem
Define a generating function
(Ng-1)-dimensional vector
W is a matrix with order (Ng-1)*(Ng-1) with entry wij=aij/di(g)
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31. Formulation of the trapping problem
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32. Formulation of the trapping problem
Setting z=1,
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33. Formulation of the trapping problem
Setting z=1,
(I-W)-1
Fundamental matrix of the Markov chain representing the unbiased random walk
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34. Formulation of the trapping problem
For large g, inverting matrix is prohibitively time and memory consuming, making it intractable to obtain MFPT through direct calculation.
Time Complexity : O(N3)
Space Complexity : O(N2)
Hence, an alternative method of computing MFPT becomes necessary.
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35. Closed-form solution to MFPT
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36. Closed-form solution to MFPT
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37. Define two generating function
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38. Closed-form solution to MFPT
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39. Conclusions
The larger the value of M, the more efficient the trapping process.
The MFPT increases as a power-law function of the number of nodes with the exponent much less than 1.
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40. Comparison
The above obtained scaling of MFPT with order of the hierarchical scale-free networks is quite different from other media.
Regular lattices
Fractals (Sierpinski, T-fractal…)
Pseudofractal (Koch, Apollonian)
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41. Analysis More Efficient The trap is fixed on hub. The modularity.
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42. Thank You
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