报告人： 林 苑 指导老师：章忠志 副教授 复旦大学 2010.10.17 第六届全国复杂网络会议 CCCN2010 Trapping in scale-free networks with hierarchical organization of modularity 报告人： 林 苑 指导老师：章忠志 副教授 复旦大学 2010.10.17 23:02:24

Outline Introduction about random walks Our works Concepts Applications Our works Fixed-trap problem Multi-trap problem Hamiltonian walks Self-avoid walks 23:02:24

Random walks At any node, go to one of the neighbors of the node with equal probability. - 23:02:24

Random walks At any node, go to one of the neighbors of the node with equal probability. - 23:02:24

Random walks At any node, go to one of the neighbors of the node with equal probability. - 23:02:24

Random walks At any node, go to one of the neighbors of the node with equal probability. - 23:02:24

Random walks At any node, go to one of the neighbors of the node with equal probability. - 23:02:24

Random walks Random walks can be depicted accurately by Markov Chain. 23:02:24

Generic Approach Markov Chain Generating Function Laplacian matrix 23:02:24

Measures Mean transit time Tij Mean return time Tii Tij ≠ Tji Mean return time Tii Mean commute time Cij Cij =Tij+Tji 23:02:24

Applications PageRank of Google Cited time Semantic categorization Recommendatory System 23:02:24

Applications One major issue: How closed are two nodes? Distance between nodes 23:02:24

Applications Classical methods Based on Random Walk (or diffusion) Shortest Path Length Numbers of Paths Based on Random Walk (or diffusion) Mean transit time, Mean commute time 23:02:24

Applications The latter methods should be better, however… Calculate inverse of matrix for O(|V|) times. Need more efficient way to calculate. 23:02:24

Trapping Problem Imagine there are traps (or absorbers) on several certain vertices. 23:02:24

Trapping Problem 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. 23:02:24

Zhang Zhongzhi, Lin Yuan, et al. Physical Review E, 2009, 80: 051120. Trapping in scale-free networks with hierarchical organization of modularity, Zhang Zhongzhi, Lin Yuan, et al. Physical Review E, 2009, 80: 051120. 23:02:24

Two remarkable features 23:02:24

Two remarkable features 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 … … 23:02:24

Trapping issue Understand how the dynamical processes are influenced by the underlying topological structure. Trapping issue relevant to a variety of contexts. 23:02:24

Modular scale-free networks 23:02:24

Modular scale-free networks 23:02:24

Modular scale-free networks 23:02:24

Modular scale-free networks 23:02:24

Modular scale-free networks 23:02:24

Modular scale-free networks We denote by Hg 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. 23:02:24

Modular scale-free networks We denote by Hg the network model after g iterations. For g>1, Hg can be obtained from Hg-1 by adding M-1 replicas of Hg-1 with their external nodes being linked to the hub of original Hg-1 unit. The new hub is the hub of original Hg-1 unit. The new external nodes are composed of all the peripheral nodes of M-1 copies of Hg-1. 23:02:24

Formulation of the trapping problem Xi First-passage time (FPT) Markov chain 23:02:24

Formulation of the trapping problem Define a generating function 23:02:24

Formulation of the trapping problem Define a generating function (Ng-1)-dimensional vector W is a matrix with order (Ng-1)*(Ng-1) with entry wij=aij/di(g) 23:02:24

Formulation of the trapping problem 23:02:24

Formulation of the trapping problem Setting z=1, 23:02:24

Formulation of the trapping problem Setting z=1, (I-W)-1 Fundamental matrix of the Markov chain representing the unbiased random walk 23:02:24

Formulation of the trapping problem For large g, inverting matrix is prohibitively time and memory consuming, making it intractable to obtain MFPT through direct calculation. Time Complexity : O(N3) Space Complexity : O(N2) Hence, an alternative method of computing MFPT becomes necessary. 23:02:24

Closed-form solution to MFPT 23:02:24

Closed-form solution to MFPT 23:02:24

Define two generating function 23:02:24

Closed-form solution to MFPT 23:02:24

Conclusions 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. 23:02:24

Comparison 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) 23:02:24

Analysis More Efficient The trap is fixed on hub. The modularity. 23:02:24

Thank You 23:02:24