Upload
iokina
View
66
Download
6
Embed Size (px)
DESCRIPTION
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: [email protected] Homepage: http://homepage.fudan.edu.cn/~zhangzz/ 20 10 年 7 月 26-31 日. Brief introduction to our group. What is a random walk. Important parameter of random walks. - PowerPoint PPT Presentation
Citation preview
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
2/43 2010-06-03复旦大学
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
3/43 2010-06-03复旦大学
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
4/43 2010-06-03复旦大学
-
Random Walks on Graphs
5/43 2010-06-03复旦大学
Random Walks on Graphs
At any node, go to one of the neighbors of the node with equal probability.
-
6/43 2010-06-03复旦大学
Random Walks on Graphs
At any node, go to one of the neighbors of the node with equal probability.
-
7/43 2010-06-03复旦大学
Random Walks on Graphs
At any node, go to one of the neighbors of the node with equal probability.
-
8/43 2010-06-03复旦大学
Random Walks on Graphs
At any node, go to one of the neighbors of the node with equal probability.
-
9/43 2010-06-03复旦大学
Random Walks on Graphs
At any node, go to one of the neighbors of the node with equal probability.
-
10/43 2010-06-03复旦大学
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)
11/43 2010-06-03复旦大学
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
12/43 2010-06-03复旦大学
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”
13/43 2010-06-03复旦大学
Application to recommendation systems
IEEE Trans. Knowl. Data Eng. 19, 355 (2007)
14/43 2010-06-03复旦大学
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
15/43 2010-06-03复旦大学
Formulas for effective resistance
16/43 2010-06-03复旦大学
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
17/43 2010-06-03复旦大学
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?
18/43 2010-06-03复旦大学
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
19/43 2010-06-03复旦大学
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
20/43 2010-06-03复旦大学
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
21/43 2010-06-03复旦大学
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
22/43 2010-06-03复旦大学
Walks on modular scale-free networks
Physical Review E, 2009, 80: 051120. 生成函数方法
23/43 2010-06-03复旦大学
Walks on Koch networks
Physical Review E, 2009, 79: 061113.
Construction
24/43 2010-06-03复旦大学
Physical Review E, 2009, 79: 061113.
Walks on Koch networks
25/43 2010-06-03复旦大学
Walks in extended Koch netoworks
26/43 2010-06-03复旦大学
Walks on a fractal scale-free network
EPL (Europhysics Letters), 2009, 88: 10001.
27/43 2010-06-03复旦大学
Walks on scale-free networks with identical degree sequences
Physical Review E, 2009, 79: 031110.
28/43 2010-06-03复旦大学
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 .
29/43 2010-06-03复旦大学
Walks on exponential networks
Conclusion: MFPT depends on the location of trap. Physical Review E, 2010, 81: 016114.
30/43 2010-06-03复旦大学
Impact of trap position on MFPT in scale-free networks
Journal of Mathematical Physics, 2009, 50: 033514.
31/43 2010-06-03复旦大学
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.
32/43 2010-06-03复旦大学
Random Walks on Vicsek fractals
Physical Review E, 2010, 81:031118.
33/43 2010-06-03复旦大学
Future workWalks with multiple traps1
Quantum walks on networks2
Biased walks, e.g. walks on weighted nets3
Self-avoid walks4
Thank You!