IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v501y2018icp153-163.html
   My bibliography  Save this article

Relationships between Perron–Frobenius eigenvalue and measurements of loops in networks

Author

Listed:
  • Chen, Lei
  • Kou, Yingxin
  • Li, Zhanwu
  • Xu, An
  • Chang, Yizhe

Abstract

The Perron–Frobenius eigenvalue (PFE) is widely used as measurement of the number of loops in networks, but what exactly the relationship between the PFE and the number of loops in networks is has not been researched yet, is it strictly monotonically increasing? And what are the relationships between the PFE and other measurements of loops in networks? Such as the average loop degree of nodes, and the distribution of loop ranks. We make researches on these questions based on samples of ER random network, NW small-world network and BA scale-free network, and the results confirm that, both the number of loops in network and the average loop degree of nodes of all samples do increase with the increase of the PFE in general trend, but neither of them are strictly monotonically increasing, so the PFE is capable to be used as a rough estimative measurement of the number of loops in networks and the average loop degree of nodes. Furthermore, we find that a majority of the loop ranks of all samples obey Weibull distribution, of which the scale parameter A and the shape parameter B have approximate power-law relationships with the PFE of the samples.

Suggested Citation

  • Chen, Lei & Kou, Yingxin & Li, Zhanwu & Xu, An & Chang, Yizhe, 2018. "Relationships between Perron–Frobenius eigenvalue and measurements of loops in networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 501(C), pages 153-163.
    • Handle: RePEc:eee:phsmap:v:501:y:2018:i:c:p:153-163
    DOI: 10.1016/j.physa.2018.02.180
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437118302711
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2018.02.180?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. M. E. J. Newman & D. J. Watts, 1999. "Renormalization Group Analysis of the Small-World Network Model," Working Papers 99-04-029, Santa Fe Institute.
    2. Loreto, V. & Paladin, G. & Pasquini, M. & Vulpiani, A., 1996. "Characterization of chaos in random maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 232(1), pages 189-200.
    3. Du, Wen-Bo & Cao, Xian-Bin & Zhao, Lin & Hu, Mao-Bin, 2009. "Evolutionary games on scale-free networks with a preferential selection mechanism," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(20), pages 4509-4514.
    4. Jinlong Ma & Weizhan Han & Qing Guo & Shuai Zhang & Junfang Wang & Zhihao Wang, 2016. "Improved efficient routing strategy on two-layer complex networks," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 27(04), pages 1-16, April.
    5. J. D. Noh, 2008. "Loop statistics in complex networks," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 66(2), pages 251-257, November.
    6. Zhang, Jun & Cao, Xian-Bin & Du, Wen-Bo & Cai, Kai-Quan, 2010. "Evolution of Chinese airport network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3922-3931.
    7. Barucca, Paolo & Lillo, Fabrizio, 2016. "Disentangling bipartite and core-periphery structure in financial networks," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 244-253.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zhang, Jianhua & Xu, Xiaoming & Hong, Liu & Wang, Shuliang & Fei, Qi, 2011. "Networked analysis of the Shanghai subway network, in China," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4562-4570.
    2. Li, Yumeng & Zhang, Jun & Perc, Matjaž, 2018. "Effects of compassion on the evolution of cooperation in spatial social dilemmas," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 437-443.
    3. Tanimoto, Jun & Nakata, Makoto & Hagishima, Aya & Ikegaya, Naoki, 2012. "Spatially correlated heterogeneous aspirations to enhance network reciprocity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 680-685.
    4. Kobayashi, Teruyoshi & Takaguchi, Taro, 2018. "Identifying relationship lending in the interbank market: A network approach," Journal of Banking & Finance, Elsevier, vol. 97(C), pages 20-36.
    5. Ping Zhu & Guiyi Wei, 2014. "Stochastic Heterogeneous Interaction Promotes Cooperation in Spatial Prisoner's Dilemma Game," PLOS ONE, Public Library of Science, vol. 9(4), pages 1-10, April.
    6. Marcus Berliant & Axel H. Watanabe, 2018. "A scale‐free transportation network explains the city‐size distribution," Quantitative Economics, Econometric Society, vol. 9(3), pages 1419-1451, November.
    7. Gong, Qiang & Wang, Kun & Fan, Xingli & Fu, Xiaowen & Xiao, Yi-bin, 2018. "International trade drivers and freight network analysis - The case of the Chinese air cargo sector," Journal of Transport Geography, Elsevier, vol. 71(C), pages 253-262.
    8. An, Sufang & Gao, Xiangyun & An, Haizhong & Liu, Siyao & Sun, Qingru & Jia, Nanfei, 2020. "Dynamic volatility spillovers among bulk mineral commodities: A network method," Resources Policy, Elsevier, vol. 66(C).
    9. Xiangyun Gao & Haizhong An & Weiqiong Zhong, 2013. "Features of the Correlation Structure of Price Indices," PLOS ONE, Public Library of Science, vol. 8(4), pages 1-9, April.
    10. Lordan, Oriol & Sallan, Jose M., 2019. "Core and critical cities of global region airport networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 724-733.
    11. Chung-Yuan Huang & Chuen-Tsai Sun & Hsun-Cheng Lin, 2005. "Influence of Local Information on Social Simulations in Small-World Network Models," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 8(4), pages 1-8.
    12. Fabio Caccioli & Paolo Barucca & Teruyoshi Kobayashi, 2018. "Network models of financial systemic risk: a review," Journal of Computational Social Science, Springer, vol. 1(1), pages 81-114, January.
    13. Zhang, Yaping & Peng, Ting & Fu, Chuanyun & Cheng, Shaowu, 2016. "Simulation analysis of factors affecting air route connection in China," Journal of Air Transport Management, Elsevier, vol. 50(C), pages 12-20.
    14. Yu, Fengyuan & Wang, Jianwei & Chen, Wei & He, Jialu, 2023. "Increased cooperation potential and risk under suppressed strategy differentiation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 621(C).
    15. Thorben Funke & Till Becker, 2019. "Stochastic block models: A comparison of variants and inference methods," PLOS ONE, Public Library of Science, vol. 14(4), pages 1-40, April.
    16. Zhang, Lan & Huang, Changwei, 2023. "Preferential selection to promote cooperation on degree–degree correlation networks in spatial snowdrift games," Applied Mathematics and Computation, Elsevier, vol. 454(C).
    17. Yu, Haitao & Guo, Xinmeng & Wang, Jiang & Deng, Bin & Wei, Xile, 2015. "Vibrational resonance in adaptive small-world neuronal networks with spike-timing-dependent plasticity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 170-179.
    18. Mark Newman, 1999. "Small Worlds: The Structure of Social Networks," Working Papers 99-12-080, Santa Fe Institute.
    19. Marnix Van Soom & Milan Van Den Heuvel & Jan Ryckebusch & Koen Schoors, 2019. "Loan Maturity Aggregation In Interbank Lending Networks Obscures Mesoscale Structure And Economic Functions," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 19/952, Ghent University, Faculty of Economics and Business Administration.
    20. An, Haizhong & Gao, Xiangyun & Fang, Wei & Ding, Yinghui & Zhong, Weiqiong, 2014. "Research on patterns in the fluctuation of the co-movement between crude oil futures and spot prices: A complex network approach," Applied Energy, Elsevier, vol. 136(C), pages 1067-1075.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:501:y:2018:i:c:p:153-163. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.