Random walks in complex networks 第第第第第第第第第第第第第第第第第第第第第第第第 章 章 章 章章章章章章章章章章章章 Email: [email protected] Homepage: http://homepage.fudan.edu.cn/~zhangzz/ 2010 章 7 章 26-31 章
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: [email protected]: 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!