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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 2010.7.30. Trapping in scale-free networks with hierarchical organization of modularity. Introduction about random walks Concepts Applications Our works Fixed-trap problem Multi-trap problem Hamiltonian walks Self-avoid walks. Outline.

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Trapping in scale-free networks with hierarchical organization of modularity

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  1. 报告人： 林 苑 指导老师：章忠志 副教授 复旦大学 2010.7.30 Trapping in scale-free networks with hierarchical organization of modularity

  2. Introduction about random walks • Concepts • Applications • Our works • Fixed-trap problem • Multi-trap problem • Hamiltonian walks • Self-avoid walks Outline

  3. Introduction about random walks • Concepts • Applications • Our works • Fixed-trap problem • Multi-trap problem • Hamiltonian walks • Self-avoid walks Outline

  4. At any node, go to one of the neighbors of the node with equal probability. Random walks -

  5. At any node, go to one of the neighbors of the node with equal probability. Random walks -

  6. Random walks • At any node, go to one of the neighbors of the node with equal probability. -

  7. Random walks • At any node, go to one of the neighbors of the node with equal probability. -

  8. Random walks • At any node, go to one of the neighbors of the node with equal probability. -

  9. Random walks • At any node, go to one of the neighbors of the node with equal probability. -

  10. Random walks can be depicted accurately by Markov Chain. Random walks

  11. Markov Chain • Laplacian matrix • Generating Function Generic Approach

  12. Mean transit time Tij • Tij ≠ Tji • Mean return time Tii • Mean commute time Cij • Cij =Tij+Tji Measures

  13. PageRank of Google Cited time Semantic categorization Recommendatory System Applications

  14. One major issue: How closed are two nodes? • Distance between nodes Applications

  15. Classical methods • Shortest Path Length • Numbers of Paths • Based on Random Walk (or diffusion) • Mean transit time, • Mean commute time Applications

  16. The latter methods should be better, however… • Calculate inverse of matrix for O(|V|) times. • Need more efficient way to calculate. Applications

  17. Imagine there are traps (or absorbers) on several certain vertices. Trapping Problem

  18. 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. Trapping Problem

  19. Trapping in scale-free networks with hierarchical organization of modularity, Zhang Zhongzhi, Lin Yuan, et al. Physical Review E, 2009, 80: 051120.

  20. Two remarkable features

  21. 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 • … … Two remarkable features

  22. Lead to the rising research on some outstanding issues in the field of complex networks such as exploring the generation mechanisms for scale-free behavior, detecting and characterizing modular structure. The two features are closely related to other structural properties such as average path length and clustering coefficient. Two remarkable features

  23. Understand how the dynamical processes are influenced by the underlying topological structure. Trapping issue relevant to a variety of contexts. Trapping issue

  24. Modular scale-free networks

  25. Modular scale-free networks

  26. Modular scale-free networks

  27. Modular scale-free networks

  28. Modular scale-free networks

  29. 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. Modular scale-free networks

  30. 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-1unit. • The new hub is the hub of original Hg-1unit. • The new external nodes are composed of all the peripheral nodes of M-1 copies of Hg-1. Modular scale-free networks

  31. Xi • First-passage time (FPT) • Markov chain Formulation of the trapping problem

  32. Define a generating function Formulation of the trapping problem

  33. Define a generating function • (Ng-1)-dimensional vector • W is a matrix with order (Ng-1)*(Ng-1) with entry wij=aij/di(g) Formulation of the trapping problem

  34. Formulation of the trapping problem

  35. Setting z=1, Formulation of the trapping problem

  36. Setting z=1, • (I-W)-1 • Fundamental matrix of the Markov chain representing the unbiased random walk Formulation of the trapping problem

  37. 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. Formulation of the trapping problem

  38. Closed-form solution to MFPT

  39. Closed-form solution to MFPT

  40. Define two generating function

  41. Closed-form solution to MFPT

  42. 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. Conclusions

  43. 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) Comparison

  44. More Efficient • The trap is fixed on hub. • The modularity. Analysis

  45. [1] Zhang Zhongzhi, Lin Yuan, et al. Trapping in scale free networks with hierarchical organization of modularity, Physical Review E, 2009, 80: 051120. [2] Zhang Zhongzhi, Lin Yuan, et al. Mean first-passage time for random walks on the T-graph, New Journal of Physics, 2009, 11: 103043. [3] Zhang Zhongzhi, Lin Yuan, et al. Average distance in a hierarchical scale-free network: an exact solution. Journal of Statistical Mechanics: Theory and Experiment, 2009, P10022. [4] Lin Yuan, Zhang Zhongzhi. Exactly determining mean first-passage time on a class of treelike regular fractals, Physical Review E, (under review). [5] Zhang Zhongzhi, Lin Yuan. Random walks in modular scale-free networks with multiple traps, Physical Review E, (in revision). [6] Zhang Zhongzhi, Lin Yuan. Impact of trap position on the efficiency of trapping in a class of dendritic scale-free networks, Journal of Chemical Physics, (under review). [7] Zhang Zhongzhi, Lin Yuan. Scaling behavior of mean first-passage time for trapping on a class of scale-free trees, European Physical Journal B, (under review). Publication

  46. Thank You

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