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Analysis of Uneven Distribution of Nodes Creating a Percolation Channel in Matrices with Translational Symmetry for Direct Current

Author

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  • Pawel Zukowski

    (Department of Economics, Vincent Pol University in Lublin, 2, Choiny Str., 20-816 Lublin, Poland)

  • Pawel Okal

    (Department of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, Poland)

  • Konrad Kierczynski

    (Department of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, Poland)

  • Przemyslaw Rogalski

    (Department of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, Poland)

  • Vitalii Bondariev

    (Department of Electrical Devices and High Voltage Technology, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, 38A, Nadbystrzycka Str., 20-618 Lublin, Poland)

Abstract

In this study, the phenomenon of node percolation was tested using the Monte Carlo computer simulation method for square matrices with dimensions L = 55, 101 and 151. The number of samples for each matrix was 5 × 10 6 . The spatial distributions of the coordinates of the nodes creating the percolation channel were determined, and maps of the density distribution of these nodes were created. It has been established that in matrices with finite dimensions, an edge phenomenon occurs, consisting of a decrease in the concentration of nodes creating a percolation channel as one approaches the edge of the matrix. As the matrix dimensions increase, the intensity of this phenomenon decreases. This expands the area in which values close to the maximum occur. The length distributions of the left and right clusters of non-conducting nodes were determined for the situation when the next randomly selected node connects them and thus reaches the percolation threshold. It was found that clusters whose dimensions are close to half of the matrix dimensions are most likely to occur. The research shows that both the values of the standard deviation of the percolation threshold and the intensity of the edge phenomenon are clearly related to the dimensions of the matrices and decrease as they increase.

Suggested Citation

  • Pawel Zukowski & Pawel Okal & Konrad Kierczynski & Przemyslaw Rogalski & Vitalii Bondariev, 2023. "Analysis of Uneven Distribution of Nodes Creating a Percolation Channel in Matrices with Translational Symmetry for Direct Current," Energies, MDPI, vol. 16(22), pages 1-14, November.
    • Handle: RePEc:gam:jeners:v:16:y:2023:i:22:p:7647-:d:1282849
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    References listed on IDEAS

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    1. Wilkinson, Leland & Friendly, Michael, 2009. "The History of the Cluster Heat Map," The American Statistician, American Statistical Association, vol. 63(2), pages 179-184.
    2. Pawel Zukowski & Paweł Okal & Konrad Kierczynski & Przemyslaw Rogalski & Sebastian Borucki & Michał Kunicki & Tomasz N. Koltunowicz, 2023. "Investigations into the Influence of Matrix Dimensions and Number of Iterations on the Percolation Phenomenon for Direct Current," Energies, MDPI, vol. 16(20), pages 1-19, October.
    3. Li, Daqing & Zhang, Qiong & Zio, Enrico & Havlin, Shlomo & Kang, Rui, 2015. "Network reliability analysis based on percolation theory," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 556-562.
    4. Cristopher Moore & M. E. J. Newman, 2000. "Epidemics and Percolation in Small-World Networks," Working Papers 00-01-002, Santa Fe Institute.
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