# 报告人： 林 苑 指导老师：章忠志 副教授 复旦大学 2010.7.30.  Introduction about random walks  Concepts  Applications  Our works  Fixed-trap problem  Multi-trap problem.

## Presentation on theme: "报告人： 林 苑 指导老师：章忠志 副教授 复旦大学 2010.7.30.  Introduction about random walks  Concepts  Applications  Our works  Fixed-trap problem  Multi-trap problem."— Presentation transcript:

 Introduction about random walks  Concepts  Applications  Our works  Fixed-trap problem  Multi-trap problem  Hamiltonian walks  Self-avoid walks

 Introduction about random walks  Concepts  Applications  Our works  Fixed-trap problem  Multi-trap problem  Hamiltonian walks  Self-avoid walks

 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.

 Markov Chain  Laplacian matrix  Generating Function

 Mean transit timeT ij  T ij ≠ T ji  Mean return timeT ii  Mean commute timeC ij  C ij =T ij +T ji

 PageRank of Google  Cited time  Semantic categorization  Recommendatory System

 One major issue: How closed are two nodes?  Distance between nodes

 Classical methods  Shortest Path Length  Numbers of Paths  Based on Random Walk (or diffusion)  Mean transit time,  Mean commute time

 The latter methods should be better, however…  Calculate inverse of matrix for O(|V|) times.  Need more efficient way to calculate.

 Imagine there are traps (or absorbers) on several certain vertices.

 We are interested the time of absorption.  For simplicity, we first consider the problem that only a single trap.

 Trapping in scale-free networks with hierarchical organization of modularity, Zhang Zhongzhi, Lin Yuan, et al. Physical Review E, 2009, 80: 051120.

 Scale-free 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  … …

 Lead to the rising research on some outstanding issues in the field of complex networks such as exploring the generation mechanisms for scale-free behavior, detecting and characterizing modular structure.  The two features are closely related to other structural properties such as average path length and clustering coefficient.

 Understand how the dynamical processes are influenced by the underlying topological structure.  Trapping issue relevant to a variety of contexts.

 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 M-1 peripheral (external) nodes. All these M nodes are fully connected to each other.

 We denote by H g the network model after g iterations.  For g>1,  H g can be obtained from H g-1 by adding M-1 replicas of H g -1 with their external nodes being linked to the hub of original H g-1 unit.  The new hub is the hub of original H g-1 unit.  The new external nodes are composed of all the peripheral nodes of M-1 copies of H g-1.

 X i  First-passage time (FPT)  Markov chain

 Define a generating function

 (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)

 Setting z=1,

 (I-W) -1  Fundamental matrix of the Markov chain representing the unbiased random walk

 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.

 The larger the value of M, the more efficient the trapping process.  The MFPT increases as a power-law function of the number of nodes with the exponent much less than 1.

 The above obtained scaling of MFPT with order of the hierarchical scale-free networks is quite different from other media.  Regular lattices  Fractals (Sierpinski, T-fractal…)  Pseudofractal (Koch, Apollonian)

 More Efficient  The trap is fixed on hub.  The modularity.

[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 first-passage time for random walks on the T-graph, New Journal of Physics, 2009, 11: 103043. [3]Zhang Zhongzhi, Lin Yuan, et al. Average distance in a hierarchical scale-free network: an exact solution. Journal of Statistical Mechanics: Theory and Experiment, 2009, P10022. [4]Lin Yuan, Zhang Zhongzhi. Exactly determining mean first-passage time on a class of treelike regular fractals, Physical Review E, (under review). [5]Zhang Zhongzhi, Lin Yuan. Random walks in modular scale-free 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 scale-free networks, Journal of Chemical Physics, (under review). [7]Zhang Zhongzhi, Lin Yuan. Scaling behavior of mean first-passage time for trapping on a class of scale-free trees, European Physical Journal B, (under review).

Thank You

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