具有模块化层次组织的无标度网络中的陷阱
Trapping in scale-free networks with hierarchical organization of modularity
摘要：
各种现实生活中的网络都具有两个显着的一般拓扑性质：无标度行为和模块化组织，研究这两个特征如何影响此类网络上发生的动态过程是自然且重要的。在本文中，我们研究了一个简单的随机过程-陷阱问题，一个完美的陷阱固定在一个给定的位置，在一个家庭的层次网络，同时表现出惊人的无标度和模块化结构的随机行走。我们专注于一个特定的情况下，与不动的陷阱定位在枢纽节点具有最大的程度。使用一种基于生成函数的方法，我们明确地确定平均首次通过时间（MFPT）的陷阱问题，这是平均的节点到陷阱的首次通过时间在整个网络。通过由分层网络的特殊结构导出的递推关系，计算出了MFPT的精确表达式。得到的严格的公式证实了广泛的直接数值计算表明，MFPT增长代数与网络的顺序。具体地，MFPT作为指数远小于1的节点数量的幂律函数而增加。我们证明了所考虑的分层网络具有更有效的结构，通过扩散传输与其他解析可溶性介质，包括一些以前研究的无标度网络。我们认为，无标度和模块化的拓扑结构是负责高效率的陷阱过程中的层次网络。
英文摘要：
A wide variety of real-life networks share two remarkable generic topological properties: scale-free behavior and modular organization, and it is natural and important to study how these two features affect the dynamical processes taking place on such networks. In this paper, we investigate a simple stochastic process--trapping problem, a random walk with a perfect trap fixed at a given location, performed on a family of hierarchical networks that exhibit simultaneously striking scale-free and modular structure. We focus on a particular case with the immobile trap positioned at the hub node having the largest degree. Using a method based on generating functions, we determine explicitly the mean first-passage time (MFPT) for the trapping problem, which is the mean of the node-to-trap first-passage time over the entire network. The exact expression for the MFPT is calculated through the recurrence relations derived from the special construction of the hierarchical networks. The obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the MFPT grows algebraically with the network order. Concretely, the MFPT increases as a power-law function of the number of nodes with the exponent much less than 1. We demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale-free networks. We argue that the scale-free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks.
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