Random walks in complex networks Email zhangzzfudan edu

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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

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

Contents Brief introduction to our group What is a random walk Important parameter of

Contents 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 2 2010 -06 -03

Brief introduction to our group v Research directions: structure and dynamics in networks §

Brief introduction to our group v 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 3 2010 -06 -03

Random Walks on Graphs - 4 2010 -06 -03

Random Walks on Graphs - 4 2010 -06 -03

Random Walks on Graphs - v At any node, go to one of the

Random Walks on Graphs - v At any node, go to one of the neighbors of the node with equal probability. 5 2010 -06 -03

Random Walks on Graphs - v At any node, go to one of the

Random Walks on Graphs - v At any node, go to one of the neighbors of the node with equal probability. 6 2010 -06 -03

Random Walks on Graphs - v At any node, go to one of the

Random Walks on Graphs - v At any node, go to one of the neighbors of the node with equal probability. 7 2010 -06 -03

Random Walks on Graphs - v At any node, go to one of the

Random Walks on Graphs - v At any node, go to one of the neighbors of the node with equal probability. 8 2010 -06 -03

Random Walks on Graphs - v At any node, go to one of the

Random Walks on Graphs - v At any node, go to one of the neighbors of the node with equal probability. 9 2010 -06 -03

Important parameters of random walks First-Passage Time F(s, t): Expected number of steps to

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

Applications of random walks v Page. Rank algorithm v Community detection v Recommendation systems

Applications of random walks v Page. Rank algorithm v Community detection v Recommendation systems v Electrical circuits (resistances) v Information Retrieval v Natural Language Processing v Machine Learning v Graph partitioning v In economics: random walk hypothesis 11 2010 -06 -03

Application to Community detection v. World Wide Web v. Citation networks v. Social networks

Application to Community detection v. World Wide Web v. Citation networks v. Social networks v. Biological networks v. Food Webs Properties of community may be quite different from the average property of network. More links “inside” than “outside” 12 2010 -06 -03

Application to recommendation systems IEEE Trans. Knowl. Data Eng. 19, 355 (2007) 13 2010

Application to recommendation systems IEEE Trans. Knowl. Data Eng. 19, 355 (2007) 13 2010 -06 -03

Connections with electrical networks v Every edge – a resistor of 1 ohm. v

Connections with electrical networks v Every edge – a resistor of 1 ohm. v 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) =2 m. R(u, v), dz is degree of z, m is the number of edges 14 2010 -06 -03

Formulas for effective resistance 15 2010 -06 -03

Formulas for effective resistance 15 2010 -06 -03

Random walks and other topologies v. Communtity structure v. Spanning trees v. Average distance

Random walks and other topologies v. Communtity structure v. Spanning trees v. Average distance EPL (Europhysics Letters), 2010, 90: 68002 16 2010 -06 -03

Our work: Random walks on complex networks with an immobile trap v Consider again

Our work: Random walks on complex networks with an immobile trap v Consider again a random walk process in a network. v In a communication or a social network, a message can disappear; for example, due to failure. v How long will the message survive before being trapped? 17 2010 -06 -03

Our work v. Random walks on scale-free networks § § § A pseudofractal scale-free

Our work v. 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 v. Random walks on exponential networks v. Random walks on fractals 18 2010 -06 -03

Main contributions v Methods for finding Mean first-passage time (MFPT) § § Backward equations

Main contributions v Methods for finding Mean first-passage time (MFPT) § § Backward equations Generating functions Laplacian spectra Electrical networks v Uncover the impacts of structures on MFPT § § Scale-free behavior Tree-like structure Modular structure Fractal structure 19 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 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 20 2010 -06 -03

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. 21 2010 -06 -03

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. 21 2010 -06 -03

Walks on modular scale-free networks 生成函数方法 Physical Review E, 2009, 80: 051120. 22 2010

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

Walks on Koch networks Construction Physical Review E, 2009, 79: 061113. 23 2010 -06

Walks on Koch networks Construction Physical Review E, 2009, 79: 061113. 23 2010 -06 -03

Walks on Koch networks Physical Review E, 2009, 79: 061113. 24 2010 -06 -03

Walks on Koch networks Physical Review E, 2009, 79: 061113. 24 2010 -06 -03

Walks in extended Koch netoworks 25 2010 -06 -03

Walks in extended Koch netoworks 25 2010 -06 -03

Walks on a fractal scale-free network EPL (Europhysics Letters), 2009, 88: 10001. 26 2010

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

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

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

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,

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 28 2010 -06 -03

Walks on exponential networks Conclusion: MFPT depends on the location of trap. Physical Review

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

Impact of trap position on MFPT in scale-free networks Journal of Mathematical Physics, 2009,

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

No qualitative effect of trap location on MFPT in the T-graph E. Agliari, Physical

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. 31 2010 -06 -03

Random Walks on Vicsek fractals Physical Review E, 2010, 81: 031118. 32 2010 -06

Random Walks on Vicsek fractals Physical Review E, 2010, 81: 031118. 32 2010 -06 -03

Future work 1 Walks with multiple traps 2 Quantum walks on networks 3 Biased

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

Thank You!

Thank You!