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Topological properties of integer networks

Author

Listed:
  • Zhou, Tao
  • Wang, Bing-Hong
  • Hui, P.M.
  • Chan, K.P.

Abstract

Inspired by Pythagoras's belief that numbers represent the reality, we study the topological properties of networks of composite numbers, in which the vertices represent the numbers and two vertices are connected if and only if there exists a divisibility relation between them. The network has a fairly large clustering coefficient C≈0.34, which is insensitive to the size of the network. The average distance between two nodes is shown to have an upper bound that is independent of the size of the network, in contrast to the behavior in small-world and ultra-small-world networks. The out-degree distribution is shown to follow a power-law behavior of the form k-2. In addition, these networks possess hierarchical structure as C(k)∼k-1 in accord with the observations of many real-life networks.

Suggested Citation

  • Zhou, Tao & Wang, Bing-Hong & Hui, P.M. & Chan, K.P., 2006. "Topological properties of integer networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 613-618.
    • Handle: RePEc:eee:phsmap:v:367:y:2006:i:c:p:613-618
    DOI: 10.1016/j.physa.2005.11.011
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    Cited by:

    1. Yinhu Zhai & Jia-Bao Liu & Shaohui Wang, 2017. "Structure Properties of Koch Networks Based on Networks Dynamical Systems," Complexity, Hindawi, vol. 2017, pages 1-7, March.
    2. Pedro A. Solares-Hernández & Fernando A. Manzano & Francisco J. Pérez-Benito & J. Alberto Conejero, 2020. "Divisibility Patterns within Pascal Divisibility Networks," Mathematics, MDPI, vol. 8(2), pages 1-10, February.
    3. Li, Yinwei & Jiang, Guo-Ping & Wu, Meng & Song, Yu-Rong & Wang, Haiyan, 2021. "Undirected Congruence Model: Topological characteristics and epidemic spreading," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).

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