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## Random walks in complex networks

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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**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**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**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**Random Walks on Graphs**• At any node, go to one of the neighbors of the node with equal probability. -**Random Walks on Graphs**• At any node, go to one of the neighbors of the node with equal probability. -**Random Walks on Graphs**• At any node, go to one of the neighbors of the node with equal probability. -**Random Walks on Graphs**• At any node, go to one of the neighbors of the node with equal probability. -**Random Walks on Graphs**• At any node, go to one of the neighbors of the node with equal probability. -**重要指标**Important parameters of random walks First-Passage TimeF(s,t): Expected number of steps to reach t starting at s Mean Commute timeC(s,t): Steps from i to j, and then go backC(t,s) = F(s,t) + F(t,s) Mean Return timeT(s,s): mean time for returning to node sfor the first time after having left it Cover time, survival problity, …… New J. Phys. 7, 26 (2005)**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**Application to Community detection**• World Wide Web • Citation networks • Social networks • Biological networks • Food Webs Properties of community may be quite different from the average property of network. More links “inside” than “outside”**Application to recommendation systems**IEEE Trans. Knowl. Data Eng. 19, 355 (2007)**Connections with electrical networks**• Every 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. v _ + u C(u,v) = F(s,t) + F(t,s) =2mR(u,v), dz is degree of z, m is the number of edges**Random walks and other topologies**• Communtity structure • Spanning trees • Average distance EPL (Europhysics Letters), 2010, 90:68002**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?**Our 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 networks • Random walks on fractals**Main 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**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**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.**Walks on modular scale-free networks**生成函数方法 Physical Review E, 2009, 80: 051120.**Walks on Koch networks**Construction Physical Review E, 2009, 79: 061113.**Walks on Koch networks**Physical Review E, 2009, 79: 061113.**Walks on a fractal scale-free network**EPL (Europhysics Letters), 2009, 88: 10001.**Walks on scale-free networks with identical degree sequences**Physical Review E, 2009, 79: 031110.**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**Walks on exponential networks**Conclusion: MFPT depends on the location of trap. Physical Review E, 2010, 81: 016114.**Impact of trap position on MFPT in scale-free networks**Journal of Mathematical Physics, 2009, 50: 033514.**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.**Random Walks on Vicsek fractals**Physical Review E, 2010, 81:031118.**Walks with multiple traps**Quantum walks on networks Biased walks, e.g. walks on weighted nets Self-avoid walks 1 2 3 4 Future work