AVERAGE FERMAT DISTANCE ON VICSEK POLYGON NETWORK

Zixuan Zhao (), Yumei Xue (), Cheng Zeng, Daohua Wang () and Zhiqiang Wu ()
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Zixuan Zhao: School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China
Yumei Xue: School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China
Cheng Zeng: ��School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong Province 264003, P. R. China
Daohua Wang: ��Graduate School of Systems and Information Engineering, University of Tsukuba, Tennodai 1-1-1, Tsukuba 305-8573, Japan
Zhiqiang Wu: School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China

FRACTALS (fractals), 2023, vol. 31, issue 09, 1-11

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.

Keywords: Vicsek Polygon; Average Fermat Distance; Self-Similar; Average Geodesic Distance; Small-World (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23501177

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