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  1. 第六届全国网络科学论坛与第二届全国混沌应用研讨会第六届全国网络科学论坛与第二届全国混沌应用研讨会 Random walks in complex networks 章 忠 志 复旦大学计算科学技术学院 Email: zhangzz@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/~zhangzz/ 2010年7月26-31日

  2. Brief introduction to our group What is a random walk Important parameter of random walks Applications of random walks Our work on Random walks: trapping in complex networks Contents

  3. Brief introduction to our group • Research directions:structure and dynamics in networks • Modeling networks and Structural analysis • Spectrum analysis and its application • Enumeration problems: spanning trees, perfect matching, Hamilton paths • Dynamics: Random walks, percolation

  4. Random Walks on Graphs -

  5. Random Walks on Graphs • At any node, go to one of the neighbors of the node with equal probability. -

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

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

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

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

  10. 重要指标 Important parameters of random walks First-Passage TimeF(s,t): Expected number of steps to reach t starting at s Mean Commute timeC(s,t): Steps from i to j, and then go backC(t,s) = F(s,t) + F(t,s) Mean Return timeT(s,s): mean time for returning to node sfor the first time after having left it Cover time, survival problity, …… New J. Phys. 7, 26 (2005)

  11. Applications of random walks • PageRank algorithm • Community detection • Recommendation systems • Electrical circuits (resistances) • Information Retrieval • Natural Language Processing • Machine Learning • Graph partitioning • In economics: random walk hypothesis

  12. Application to Community detection • World Wide Web • Citation 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”

  13. Application to recommendation systems IEEE Trans. Knowl. Data Eng. 19, 355 (2007)

  14. 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), dz is degree of z, m is the number of edges

  15. Formulas for effective resistance

  16. Random walks and other topologies • Communtity structure • Spanning trees • Average distance EPL (Europhysics Letters), 2010, 90:68002

  17. Our work: Random walks on complex networks with an immobile trap Consider again a random walk process in a network. In a communication or a social network, a message can disappear; for example, due to failure. How long will the message survive before being trapped?

  18. 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 • Random walks on exponential networks • Random walks on fractals

  19. Main contributions • 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 • Tree-like structure • Modular structure • Fractal structure

  20. Walks on pseudofractal scale-free web Physical Review E, 2009, 79: 021127. 主要贡献:(1)新的解析方法 (2)新发现

  21. Walks on Apollonian network 为发表时所报导的传输效率最高的网络 EPL, 2009, 86: 10006.

  22. Walks on modular scale-free networks 生成函数方法 Physical Review E, 2009, 80: 051120.

  23. Walks on Koch networks Construction Physical Review E, 2009, 79: 061113.

  24. Walks on Koch networks Physical Review E, 2009, 79: 061113.

  25. Walksin extended Koch netoworks

  26. Walks on a fractal scale-free network EPL (Europhysics Letters), 2009, 88: 10001.

  27. Walks on scale-free networks with identical degree sequences Physical Review E, 2009, 79: 031110.

  28. Walks on scale-free networks with identical degree sequences 模型优点:(1) 不需要交叉边;(2)网络始终连通. Physical Review E, 2009, 80: 061111

  29. Walks on exponential networks Conclusion: MFPT depends on the location of trap. Physical Review E, 2010, 81: 016114.

  30. Impact of trap position on MFPT in scale-free networks Journal of Mathematical Physics, 2009, 50: 033514.

  31. No qualitative effect of trap location on MFPT in the T-graph E. Agliari, Physical Review E, 2008, 77: 011128. Zhang ZZ, et. al.,New Journal of Physics, 2009, 11: 103043.

  32. Random Walks on Vicsek fractals Physical Review E, 2010, 81:031118.

  33. Walks with multiple traps Quantum walks on networks Biased walks, e.g. walks on weighted nets Self-avoid walks 1 2 3 4 Future work

  34. Thank You!