Random walks in complex networks

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

ContentsContents

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Brief introduction to our groupBrief 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 Random Walks on Graphs

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Random Walks on Graphs 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 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 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 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 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 walksImportant 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 walksApplications 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 detectionApplication 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 systemsApplication to recommendation systems

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

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Connections with electrical networksConnections 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 zz

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

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Formulas for effective resistanceFormulas for effective resistance

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Random walks and other topologiesRandom walks and other topologies

Communtity 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

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 workOur 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 networksRandom walks on fractals

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Main contributionsMain 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

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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 networkWalks 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 networksWalks 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 Walks in extended Koch netoworks

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Walks on a fractal scale-free networkWalks on a fractal scale-free network

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

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Walks on scale-free networks with identical degree sequencesWalks 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 sequencesWalks 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 networksWalks 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-graphNo 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 fractalsRandom Walks on Vicsek fractals

Physical Review E, 2010, 81:031118.

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Future workFuture work

Walks with multiple traps1

Quantum walks on networks2

Biased walks, e.g. walks on weighted nets3

Self-avoid walks4

Thank You!