Average Fermat distance on Vicsek polygon network

IF 3.3 3区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Fractals-Complex Geometry Patterns and Scaling in Nature and Society Pub Date : 2023-11-01 DOI:10.1142/s0218348x23501177

Zixuan Zhao, Yumei Xue, Cheng Zeng, Daohua Wang, Zhiqiang Wu

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引用次数: 0

Abstract

The Fermat problem is a crucial topological issue corresponding to fractal networks. In this paper, we discuss the average Fermat distance (AFD) of the Vicsek polygon network and analyze structural properties. We construct the Vicsek polygon network based on Vicsek fractal in an iterative way. Given the structure of network, we present an elaborate analysis of the Fermat point under various situations. The special network structure allows a way to calculate the AFD based on average geodesic distance (AGD). Moreover, we introduce the Vicsek polygon fractal and calculate its AGD and AFD. Its relationship with the network enables us to deduce the above two indices of the network directly. The results show that both in network and fractal, the ratio of AFD and AGD tends to 3/2, which demonstrates that both of them can serve as indicators of small-world property of complex networks. In fact, in Vicsek polygon network, the AFD grows linearly with network order, implying that our evolving network does not possess the small-world property.

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维塞克多边形网络上的平均费马距离

费马问题是分形网络的一个重要拓扑问题。本文讨论了Vicsek多边形网络的平均费马距离(AFD)，并分析了其结构性质。基于维塞克分形，采用迭代的方法构造维塞克多边形网络。在给定网络结构的情况下，我们详细分析了各种情况下的费马点。特殊的网络结构允许基于平均测地线距离(AGD)计算AFD的方法。此外，我们引入了Vicsek多边形分形，并计算了它的AGD和AFD。它与网络的关系使我们可以直接推导出网络的上述两个指标。结果表明，在网络和分形中，AFD和AGD的比值都趋向于3/2，这表明两者都可以作为复杂网络小世界性质的指标。事实上，在Vicsek多边形网络中，AFD随网络阶数线性增长，这意味着我们进化的网络不具有小世界性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Fractals-Complex Geometry Patterns and Scaling in Nature and Society 数学-数学跨学科应用

CiteScore

7.40

自引率

23.40%

发文量

319

审稿时长

>12 weeks

期刊介绍： The investigation of phenomena involving complex geometry, patterns and scaling has gone through a spectacular development and applications in the past decades. For this relatively short time, geometrical and/or temporal scaling have been shown to represent the common aspects of many processes occurring in an unusually diverse range of fields including physics, mathematics, biology, chemistry, economics, engineering and technology, and human behavior. As a rule, the complex nature of a phenomenon is manifested in the underlying intricate geometry which in most of the cases can be described in terms of objects with non-integer (fractal) dimension. In other cases, the distribution of events in time or various other quantities show specific scaling behavior, thus providing a better understanding of the relevant factors determining the given processes. Using fractal geometry and scaling as a language in the related theoretical, numerical and experimental investigations, it has been possible to get a deeper insight into previously intractable problems. Among many others, a better understanding of growth phenomena, turbulence, iterative functions, colloidal aggregation, biological pattern formation, stock markets and inhomogeneous materials has emerged through the application of such concepts as scale invariance, self-affinity and multifractality. The main challenge of the journal devoted exclusively to the above kinds of phenomena lies in its interdisciplinary nature; it is our commitment to bring together the most recent developments in these fields so that a fruitful interaction of various approaches and scientific views on complex spatial and temporal behaviors in both nature and society could take place.