Role of fractal dimension in random walks on scale-free networks |
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Authors: | Zhongzhi Zhang Yihang Yang and Shuyang Gao |
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Institution: | (2) Institute for Theoretical Physics, Cologne University, K?ln, Germany; |
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Abstract: | Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal
dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are
ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic,
often called (x,y)-flowers; the other is random, which is a combination of (1,3)-flower and (2,4)-flower and thus called hybrid networks. The
two network families display rich behavior as observed in various real systems, as well as some unique topological properties
not shared by other networks. We derive analytically the average trapping time for random walks on both the (x,y)-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree
in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing
the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping
time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension. |
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