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
(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.
Zhang Zhongzhi, Lin Yuan, et al. Trapping in scale free networks with hierarchical organization of modularity, Physical Review E, 2009, 80: 051120. Zhang Zhongzhi, Lin Yuan, et al. Mean first-passage time for random walks on the T-graph, New Journal of Physics, 2009, 11: 103043. 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. Lin Yuan, Zhang Zhongzhi. Exactly determining mean first-passage time on a class of treelike regular fractals, Physical Review E, (under review). Zhang Zhongzhi, Lin Yuan. Random walks in modular scale-free networks with multiple traps, Physical Review E, (in revision). 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). 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).