Random walks in complex networks

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.cnHomepage: http://homepage.fudan.edu.cn/~zhangzz/2010 年 7 月 26-31 日

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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

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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

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Random Walks on Graphs

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Random Walks on Graphs

At any node, go to one of the neighbors of the node with equal probability.

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Random Walks on Graphs

At any node, go to one of the neighbors of the node with equal probability.

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Random Walks on Graphs

At any node, go to one of the neighbors of the node with equal probability.

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Random Walks on Graphs

At any node, go to one of the neighbors of the node with equal probability.

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Random Walks on Graphs

At any node, go to one of the neighbors of the node with equal probability.

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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)

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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

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Application to Community detection

World Wide WebCitation networksSocial networksBiological networksFood Webs

Properties of community may be quite different from the average property of network.More links “inside” than “outside”

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Application to recommendation systems

IEEE Trans. Knowl. Data Eng. 19, 355 (2007)

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Connections with electrical networksEvery 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

1( , ) ( , ) ( , ) ( , )2 z

z

F s t d R s t R t z R s z

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Formulas for effective resistance

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Random walks and other topologiesCommuntity structureSpanning treesAverage distance

2

1( )N

ST ii

N GN

( , ) ( )( , )

( )

u vST

ST

N GR u v

N G

EPL (Europhysics Letters), 2010, 90:68002

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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?

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Our workRandom 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 networksRandom walks on fractals

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Main contributionsMethods 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

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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

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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

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Walks on modular scale-free networks

Physical Review E, 2009, 80: 051120. 生成函数方法

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Walks on Koch networks

Physical Review E, 2009, 79: 061113.

Construction

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Physical Review E, 2009, 79: 061113.

Walks on Koch networks

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Walks in extended Koch netoworks

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Walks on a fractal scale-free network

EPL (Europhysics Letters), 2009, 88: 10001.

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Walks on scale-free networks with identical degree sequences

Physical Review E, 2009, 79: 031110.

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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 .

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Walks on exponential networks

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

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Impact of trap position on MFPT in scale-free networks

Journal of Mathematical Physics, 2009, 50: 033514.

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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.

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Random Walks on Vicsek fractals

Physical Review E, 2010, 81:031118.

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Future workWalks with multiple traps1

Quantum walks on networks2

Biased walks, e.g. walks on weighted nets3

Self-avoid walks4

Thank You!