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

2. 复旦大学 2/43 2010-06-03 Brief introduction to our groupWhat is a random walkImportant parameter of random walksApplications of random walksOur work on Random walks: trapping in complex networks Contents

3. 复旦大学 3/43 2010-06-03 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. 复旦大学 4/43 2010-06-03 - Random Walks on Graphs

5. 复旦大学 5/43 2010-06-03 Random Walks on Graphs  At any node, go to one of the neighbors of the node with equal probability. -

6. 复旦大学 6/43 2010-06-03 Random Walks on Graphs  At any node, go to one of the neighbors of the node with equal probability. -

7. 复旦大学 7/43 2010-06-03 Random Walks on Graphs  At any node, go to one of the neighbors of the node with equal probability. -

8. 复旦大学 8/43 2010-06-03 Random Walks on Graphs  At any node, go to one of the neighbors of the node with equal probability. -

9. 复旦大学 9/43 2010-06-03 Random Walks on Graphs  At any node, go to one of the neighbors of the node with equal probability. -

10. 复旦大学 10/43 2010-06-03 Important parameters of random walks 重要指标 Mean Commute time C(s,t): Steps from i to j, 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 First-Passage Time F(s,t): Expected number of steps to reach t starting at s Cover time, survival problity, …… New J. Phys. 7, 26 (2005)

11. 复旦大学 11/43 2010-06-03 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. 复旦大学 12/43 2010-06-03 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. 复旦大学 13/43 2010-06-03 Application to recommendation systems IEEE Trans. Knowl. Data Eng. 19, 355 (2007)

14. 复旦大学 14/43 2010-06-03 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. _ u v + C(u,v) = F(s,t) + F(t,s) =2mR(u,v), dz is degree of z, m is the number of edges

15. 复旦大学 15/43 2010-06-03 Formulas for effective resistance

16. 复旦大学 16/43 2010-06-03 Random walks and other topologies  Communtity structure  Spanning trees  Average distance EPL (Europhysics Letters), 2010, 90:68002

17. 复旦大学 17/43 2010-06-03 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. 复旦大学 18/43 2010-06-03 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. 复旦大学 19/43 2010-06-03 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. 复旦大学 20/43 2010-06-03 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. 复旦大学 21/43 2010-06-03 Walks on Apollonian network EPL, 2009, 86: 10006. 为发表时所报导的传输效率最高的网络

22. 复旦大学 22/43 2010-06-03 Walks on modular scale-free networks Physical Review E, 2009, 80: 051120. 生成函数方法

23. 复旦大学 23/43 2010-06-03 Walks on Koch networks Physical Review E, 2009, 79: 061113. Construction

24. 复旦大学 24/43 2010-06-03 Physical Review E, 2009, 79: 061113. Walks on Koch networks

25. 复旦大学 25/43 2010-06-03 Walks in extended Koch netoworks

26. 复旦大学 26/43 2010-06-03 Walks on a fractal scale-free network EPL (Europhysics Letters), 2009, 88: 10001.

27. 复旦大学 27/43 2010-06-03 Walks on scale-free networks with identical degree sequences Physical Review E, 2009, 79: 031110.

28. 复旦大学 28/43 2010-06-03 Walks on scale-free networks with identical degree sequences Physical Review E, 2009, 80: 061111 模型优点： (1) 不需要交叉边； (2) 网络始终连通.

29. 复旦大学 29/43 2010-06-03 Walks on exponential networks Conclusion: MFPT depends on the location of trap. Physical Review E, 2010, 81: 016114.

30. 复旦大学 30/43 2010-06-03 Impact of trap position on MFPT in scale-free networks Journal of Mathematical Physics, 2009, 50: 033514.

31. 复旦大学 31/43 2010-06-03 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. 复旦大学 32/43 2010-06-03 Random Walks on Vicsek fractals Physical Review E, 2010, 81:031118.

33. 复旦大学 33/43 2010-06-03 Future work Walks with multiple traps 1 Quantum walks on networks 2 Biased walks, e.g. walks on weighted nets 3 Self-avoid walks 4

34. Thank You!