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The maximum number of 3- and 4-cliques within a planar maximally filtered graph

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  • Birch, Jenna
  • Pantelous, Athanasios A.
  • Zuev, Konstantin

Abstract

Planar Maximally Filtered Graphs (PMFG) are an important tool for filtering the most relevant information from correlation based networks such as stock market networks. One of the main characteristics of a PMFG is the number of its 3- and 4-cliques. Recently in a few high impact papers it was stated that, based on heuristic evidence, the maximum number of 3- and 4-cliques that can exist in a PMFG with n vertices is 3n−8 and n−4 respectively. In this paper, we prove that this is indeed the case.

Suggested Citation

  • Birch, Jenna & Pantelous, Athanasios A. & Zuev, Konstantin, 2015. "The maximum number of 3- and 4-cliques within a planar maximally filtered graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 221-229.
    • Handle: RePEc:eee:phsmap:v:417:y:2015:i:c:p:221-229
    DOI: 10.1016/j.physa.2014.09.011
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    References listed on IDEAS

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    1. Mantegna,Rosario N. & Stanley,H. Eugene, 2007. "Introduction to Econophysics," Cambridge Books, Cambridge University Press, number 9780521039871, October.
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    Cited by:

    1. Jenna Birch & Athanasios A. Pantelous & Kimmo Soramäki, 2016. "Analysis of Correlation Based Networks Representing DAX 30 Stock Price Returns," Computational Economics, Springer;Society for Computational Economics, vol. 47(4), pages 501-525, April.
    2. Chen, Yanhua & Li, Youwei & Pantelous, Athanasios A. & Stanley, H. Eugene, 2022. "Short-run disequilibrium adjustment and long-run equilibrium in the international stock markets: A network-based approach," International Review of Financial Analysis, Elsevier, vol. 79(C).
    3. Yong Tang & Jason Jie Xiong & Zi-Yang Jia & Yi-Cheng Zhang, 2018. "Complexities in Financial Network Topological Dynamics: Modeling of Emerging and Developed Stock Markets," Complexity, Hindawi, vol. 2018, pages 1-31, November.
    4. Vamvakaris, Michail D. & Pantelous, Athanasios A. & Zuev, Konstantin M., 2018. "Time series analysis of S&P 500 index: A horizontal visibility graph approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 497(C), pages 41-51.
    5. Guo, Xue & Li, Weibo & Zhang, Hu & Tian, Tianhai, 2022. "Multi-likelihood methods for developing relationship networks using stock market data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 585(C).

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