控制树状分形中的捕获效率
Controlling the efficiency of trapping in treelike fractals
摘要：
有效控制捕获过程，特别是捕获效率，是复杂系统中捕获问题研究的核心，因为它是各种其他动态过程的基本机制。因此，研究陷阱问题的控制技术具有重要的理论和现实意义。在本文中，我们研究了一族在中心节点处具有深陷阱的有向分形中的陷阱问题。有向分形是对无向分形的推广，它引入了由参数支配的有向边权。我们描述了所有的特征值和它们的退化为一个相关的矩阵控制的捕获过程。特征值通过从分形的自相似结构推导出的精确递归关系提供。我们还获得了最小特征值和平均首次通过时间（MFPT）作为捕获效率的衡量标准，这是步行者第一次访问陷阱的预期时间的表达式。根据已证明的近似等于最小特征值的倒数的事实，对MFPT进行了评价。我们证明了MFPT是由权参数控制的，通过修改权参数，MFPT可以随系统大小超线性、线性或次线性地扩展。因此，这项工作为精细控制分形中的捕获过程铺平了道路。
英文摘要：
Efficiently controlling the trapping process, especially the trapping efficiency, is central in the study of trap problem in complex systems, since it is a fundamental mechanism for diverse other dynamic processes. Thus, it is of theoretical and practical significance to study the control technique for trapping problem. In this paper, we study the trapping problem in a family of proposed directed fractals with a deep trap at a central node. The directed fractals are a generalization of previous undirected fractals by introducing the directed edge weights dominated by a parameter. We characterize all the eigenvalues and their degeneracies for an associated matrix governing the trapping process. The eigenvalues are provided through an exact recursive relation deduced from the self-similar structure of the fractals. We also obtain the expressions for the smallest eigenvalue and the mean first-passage time (MFPT) as a measure of trapping efficiency, which is the expected time for the walker to first visit the trap. The MFPT is evaluated according to the proved fact that it is approximately equal to reciprocal of the smallest eigenvalue. We show that the MFPT is controlled by the weight parameter, by modifying which, the MFPT can scale superlinealy, linearly, or sublinearly with the system size. Thus, this work paves a way to delicately controlling the trapping process in the fractals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
