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报告人： 林 苑 指导老师：章忠志 副教授 复旦大学 2010.7.30
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Introduction about random walks Concepts Applications Our works Fixedtrap problem Multitrap problem Hamiltonian walks Selfavoid walks
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Introduction about random walks Concepts Applications Our works Fixedtrap problem Multitrap problem Hamiltonian walks Selfavoid walks
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At any node, go to one of the neighbors of the node with equal probability. 
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 At any node, go to one of the neighbors of the node with equal probability.
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Random walks can be depicted accurately by Markov Chain.
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Markov Chain Laplacian matrix Generating Function
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Mean transit timeT ij T ij ≠ T ji Mean return timeT ii Mean commute timeC ij C ij =T ij +T ji
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PageRank of Google Cited time Semantic categorization Recommendatory System
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One major issue: How closed are two nodes? Distance between nodes
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Classical methods Shortest Path Length Numbers of Paths Based on Random Walk (or diffusion) Mean transit time, Mean commute time
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The latter methods should be better, however… Calculate inverse of matrix for O(V) times. Need more efficient way to calculate.
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Imagine there are traps (or absorbers) on several certain vertices.
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We are interested the time of absorption. For simplicity, we first consider the problem that only a single trap.
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Trapping in scalefree networks with hierarchical organization of modularity, Zhang Zhongzhi, Lin Yuan, et al. Physical Review E, 2009, 80: 051120.
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Scalefree topology Modular organization For a large number of real networks, these two features coexist: Protein interaction network Metabolic networks The World Wide Web Some social networks … …
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Lead to the rising research on some outstanding issues in the field of complex networks such as exploring the generation mechanisms for scalefree behavior, detecting and characterizing modular structure. The two features are closely related to other structural properties such as average path length and clustering coefficient.
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Understand how the dynamical processes are influenced by the underlying topological structure. Trapping issue relevant to a variety of contexts.
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We denote by H g the network model after g iterations. For g=1, The network consists of a central node, called the hub node, And M1 peripheral (external) nodes. All these M nodes are fully connected to each other.
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We denote by H g the network model after g iterations. For g>1, H g can be obtained from H g1 by adding M1 replicas of H g 1 with their external nodes being linked to the hub of original H g1 unit. The new hub is the hub of original H g1 unit. The new external nodes are composed of all the peripheral nodes of M1 copies of H g1.
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X i Firstpassage time (FPT) Markov chain
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Define a generating function
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(N g 1)dimensional vector W is a matrix with order (N g 1)*(N g 1) with entry w ij =a ij /d i (g)
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Setting z=1,
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(IW) 1 Fundamental matrix of the Markov chain representing the unbiased random walk
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For large g, inverting matrix is prohibitively time and memory consuming, making it intractable to obtain MFPT through direct calculation. Time Complexity : O(N 3 ) Space Complexity : O(N 2 ) Hence, an alternative method of computing MFPT becomes necessary.
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The larger the value of M, the more efficient the trapping process. The MFPT increases as a powerlaw function of the number of nodes with the exponent much less than 1.
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The above obtained scaling of MFPT with order of the hierarchical scalefree networks is quite different from other media. Regular lattices Fractals (Sierpinski, Tfractal…) Pseudofractal (Koch, Apollonian)
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More Efficient The trap is fixed on hub. The modularity.
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[1]Zhang Zhongzhi, Lin Yuan, et al. Trapping in scale free networks with hierarchical organization of modularity, Physical Review E, 2009, 80: 051120. [2]Zhang Zhongzhi, Lin Yuan, et al. Mean firstpassage time for random walks on the Tgraph, New Journal of Physics, 2009, 11: 103043. [3]Zhang Zhongzhi, Lin Yuan, et al. Average distance in a hierarchical scalefree network: an exact solution. Journal of Statistical Mechanics: Theory and Experiment, 2009, P10022. [4]Lin Yuan, Zhang Zhongzhi. Exactly determining mean firstpassage time on a class of treelike regular fractals, Physical Review E, (under review). [5]Zhang Zhongzhi, Lin Yuan. Random walks in modular scalefree networks with multiple traps, Physical Review E, (in revision). [6]Zhang Zhongzhi, Lin Yuan. Impact of trap position on the efficiency of trapping in a class of dendritic scalefree networks, Journal of Chemical Physics, (under review). [7]Zhang Zhongzhi, Lin Yuan. Scaling behavior of mean firstpassage time for trapping on a class of scalefree trees, European Physical Journal B, (under review).
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Thank You
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