Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks

Dandan Ye; Fei Zhang; Yiteng Qin; Xiaojuan Zhang; Ning Zhang; Jin Qin; Wei Chen; Yingze Zhang

doi:doi:10.11648/j.acm.20231205.12

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Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks

Dandan Ye, Fei Zhang, Yiteng Qin, Xiaojuan Zhang, Ning Zhang, Jin Qin, Wei Chen, Yingze Zhang

Published in Applied and Computational Mathematics (Volume 12, Issue 5)

Received: 31 August 2023 Accepted: 25 September 2023 Published: 1 November 2023

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Abstract

The weighted extended Cayley networks are an extension of extended Cayley networks, which are the structures constructed by introducing power spaces into traditional Cayley trees. The weighted extended Cayley networks are constructed depending on two structural parameters of the network m, n and a weight factor r. Firstly, we used a new calculation method to calculate the exact analytic formula of the average weighted shortest path (AWSP). The obtained results show that: (1) For very large systems, the AWSPs for different value of weight factor r are less affected by the parameter m. (2) The AWSPs are less affected by the weight factor r when r is greater than 0 and less than or equal to n, while the AWSPs depend on the scaling factor r when r is greater than n. We have presented a trapping issue of weight-dependent walks in the weighted extended Cayley networks, focusing on a specific case with a perfect trap located at the central node. Then, the scaling expression of the average trapping time (ATT) is derived based on the layering of weighted extended Cayley networks. It was surprisingly found that (1) Regardless of the relationship between m and n, the dominant terms of ATTs are consistent. (2) ATTs are less affected by the structural parameter m and the weight factor r when r is less than or equal to the ratio of n to m−1, indicating that the efficiency of the trapping process is independent of m and r. (3) When r is greater than the ratio of n to m−1, the efficiency of the trapping process depends on three main parameters: two structural parameters of the networks m, n and a weight factor r, which means that the smaller the multiplier of three numbers r, n and m − 1 is, the more efficient the trapping process is. Therefore, the trapping efficiency of the weighted extended Cayley networks is not only affected by the underlying structures of the networks m and n, but also by the weight factor r.

Published in	Applied and Computational Mathematics (Volume 12, Issue 5)
DOI	10.11648/j.acm.20231205.12
Page(s)	114-139
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Trapping Time, Average Weighted Shortest Path, Weighted Extended Cayley Networks, Weight-dependent Walk

References

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APA Style

Dandan Ye, Fei Zhang, Yiteng Qin, Xiaojuan Zhang, Ning Zhang, et al. (2023). Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks. Applied and Computational Mathematics, 12(5), 114-139. https://doi.org/10.11648/j.acm.20231205.12

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ACS Style

Dandan Ye; Fei Zhang; Yiteng Qin; Xiaojuan Zhang; Ning Zhang, et al. Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks. Appl. Comput. Math. 2023, 12(5), 114-139. doi: 10.11648/j.acm.20231205.12

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AMA Style

Dandan Ye, Fei Zhang, Yiteng Qin, Xiaojuan Zhang, Ning Zhang, et al. Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks. Appl Comput Math. 2023;12(5):114-139. doi: 10.11648/j.acm.20231205.12

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@article{10.11648/j.acm.20231205.12,
  author = {Dandan Ye and Fei Zhang and Yiteng Qin and Xiaojuan Zhang and Ning Zhang and Jin Qin and Wei Chen and Yingze Zhang},
  title = {Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {5},
  pages = {114-139},
  doi = {10.11648/j.acm.20231205.12},
  url = {https://doi.org/10.11648/j.acm.20231205.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231205.12},
  abstract = {The weighted extended Cayley networks are an extension of extended Cayley networks, which are the structures constructed by introducing power spaces into traditional Cayley trees. The weighted extended Cayley networks are constructed depending on two structural parameters of the network m, n and a weight factor r. Firstly, we used a new calculation method to calculate the exact analytic formula of the average weighted shortest path (AWSP). The obtained results show that: (1) For very large systems, the AWSPs for different value of weight factor r are less affected by the parameter m. (2) The AWSPs are less affected by the weight factor r when r is greater than 0 and less than or equal to n, while the AWSPs depend on the scaling factor r when r is greater than n. We have presented a trapping issue of weight-dependent walks in the weighted extended Cayley networks, focusing on a specific case with a perfect trap located at the central node. Then, the scaling expression of the average trapping time (ATT) is derived based on the layering of weighted extended Cayley networks. It was surprisingly found that (1) Regardless of the relationship between m and n, the dominant terms of ATTs are consistent. (2) ATTs are less affected by the structural parameter m and the weight factor r when r is less than or equal to the ratio of n to m−1, indicating that the efficiency of the trapping process is independent of m and r. (3) When r is greater than the ratio of n to m−1, the efficiency of the trapping process depends on three main parameters: two structural parameters of the networks m, n and a weight factor r, which means that the smaller the multiplier of three numbers r, n and m − 1 is, the more efficient the trapping process is. Therefore, the trapping efficiency of the weighted extended Cayley networks is not only affected by the underlying structures of the networks m and n, but also by the weight factor r.
},
 year = {2023}
}

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TY  - JOUR
T1  - Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks
AU  - Dandan Ye
AU  - Fei Zhang
AU  - Yiteng Qin
AU  - Xiaojuan Zhang
AU  - Ning Zhang
AU  - Jin Qin
AU  - Wei Chen
AU  - Yingze Zhang
Y1  - 2023/11/01
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231205.12
DO  - 10.11648/j.acm.20231205.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 114
EP  - 139
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231205.12
AB  - The weighted extended Cayley networks are an extension of extended Cayley networks, which are the structures constructed by introducing power spaces into traditional Cayley trees. The weighted extended Cayley networks are constructed depending on two structural parameters of the network m, n and a weight factor r. Firstly, we used a new calculation method to calculate the exact analytic formula of the average weighted shortest path (AWSP). The obtained results show that: (1) For very large systems, the AWSPs for different value of weight factor r are less affected by the parameter m. (2) The AWSPs are less affected by the weight factor r when r is greater than 0 and less than or equal to n, while the AWSPs depend on the scaling factor r when r is greater than n. We have presented a trapping issue of weight-dependent walks in the weighted extended Cayley networks, focusing on a specific case with a perfect trap located at the central node. Then, the scaling expression of the average trapping time (ATT) is derived based on the layering of weighted extended Cayley networks. It was surprisingly found that (1) Regardless of the relationship between m and n, the dominant terms of ATTs are consistent. (2) ATTs are less affected by the structural parameter m and the weight factor r when r is less than or equal to the ratio of n to m−1, indicating that the efficiency of the trapping process is independent of m and r. (3) When r is greater than the ratio of n to m−1, the efficiency of the trapping process depends on three main parameters: two structural parameters of the networks m, n and a weight factor r, which means that the smaller the multiplier of three numbers r, n and m − 1 is, the more efficient the trapping process is. Therefore, the trapping efficiency of the weighted extended Cayley networks is not only affected by the underlying structures of the networks m and n, but also by the weight factor r.

VL  - 12
IS  - 5
ER  -

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Author Information

Dandan Ye

Institute of Orthopaedics, the Third Hospital of Hebei Medical University, Shijiazhuang, China
Fei Zhang

Institute of Orthopaedics, the Third Hospital of Hebei Medical University, Shijiazhuang, China
Yiteng Qin

Hebei Medical University Basical Medicine College, Shijiazhuang, China
Xiaojuan Zhang

Institute of Orthopaedics, the Third Hospital of Hebei Medical University, Shijiazhuang, China
Ning Zhang

Institute of Orthopaedics, the Third Hospital of Hebei Medical University, Shijiazhuang, China
Jin Qin

Hebei Orthopedic Clinical Research Center, Shijiazhuang, China
Wei Chen

Institute of Orthopaedics, the Third Hospital of Hebei Medical University, Shijiazhuang, China
Yingze Zhang

Institute of Orthopaedics, the Third Hospital of Hebei Medical University, Shijiazhuang, China

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APA Style

Dandan Ye, Fei Zhang, Yiteng Qin, Xiaojuan Zhang, Ning Zhang, et al. (2023). Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks. Applied and Computational Mathematics, 12(5), 114-139. https://doi.org/10.11648/j.acm.20231205.12

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ACS Style

Dandan Ye; Fei Zhang; Yiteng Qin; Xiaojuan Zhang; Ning Zhang, et al. Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks. Appl. Comput. Math. 2023, 12(5), 114-139. doi: 10.11648/j.acm.20231205.12

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AMA Style

Dandan Ye, Fei Zhang, Yiteng Qin, Xiaojuan Zhang, Ning Zhang, et al. Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks. Appl Comput Math. 2023;12(5):114-139. doi: 10.11648/j.acm.20231205.12

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@article{10.11648/j.acm.20231205.12,
  author = {Dandan Ye and Fei Zhang and Yiteng Qin and Xiaojuan Zhang and Ning Zhang and Jin Qin and Wei Chen and Yingze Zhang},
  title = {Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {5},
  pages = {114-139},
  doi = {10.11648/j.acm.20231205.12},
  url = {https://doi.org/10.11648/j.acm.20231205.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231205.12},
  abstract = {The weighted extended Cayley networks are an extension of extended Cayley networks, which are the structures constructed by introducing power spaces into traditional Cayley trees. The weighted extended Cayley networks are constructed depending on two structural parameters of the network m, n and a weight factor r. Firstly, we used a new calculation method to calculate the exact analytic formula of the average weighted shortest path (AWSP). The obtained results show that: (1) For very large systems, the AWSPs for different value of weight factor r are less affected by the parameter m. (2) The AWSPs are less affected by the weight factor r when r is greater than 0 and less than or equal to n, while the AWSPs depend on the scaling factor r when r is greater than n. We have presented a trapping issue of weight-dependent walks in the weighted extended Cayley networks, focusing on a specific case with a perfect trap located at the central node. Then, the scaling expression of the average trapping time (ATT) is derived based on the layering of weighted extended Cayley networks. It was surprisingly found that (1) Regardless of the relationship between m and n, the dominant terms of ATTs are consistent. (2) ATTs are less affected by the structural parameter m and the weight factor r when r is less than or equal to the ratio of n to m−1, indicating that the efficiency of the trapping process is independent of m and r. (3) When r is greater than the ratio of n to m−1, the efficiency of the trapping process depends on three main parameters: two structural parameters of the networks m, n and a weight factor r, which means that the smaller the multiplier of three numbers r, n and m − 1 is, the more efficient the trapping process is. Therefore, the trapping efficiency of the weighted extended Cayley networks is not only affected by the underlying structures of the networks m and n, but also by the weight factor r.
},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Trapping Issues for Weight-dependent Walks in the Weighted Extended Cayley Networks
AU  - Dandan Ye
AU  - Fei Zhang
AU  - Yiteng Qin
AU  - Xiaojuan Zhang
AU  - Ning Zhang
AU  - Jin Qin
AU  - Wei Chen
AU  - Yingze Zhang
Y1  - 2023/11/01
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231205.12
DO  - 10.11648/j.acm.20231205.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 114
EP  - 139
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231205.12
AB  - The weighted extended Cayley networks are an extension of extended Cayley networks, which are the structures constructed by introducing power spaces into traditional Cayley trees. The weighted extended Cayley networks are constructed depending on two structural parameters of the network m, n and a weight factor r. Firstly, we used a new calculation method to calculate the exact analytic formula of the average weighted shortest path (AWSP). The obtained results show that: (1) For very large systems, the AWSPs for different value of weight factor r are less affected by the parameter m. (2) The AWSPs are less affected by the weight factor r when r is greater than 0 and less than or equal to n, while the AWSPs depend on the scaling factor r when r is greater than n. We have presented a trapping issue of weight-dependent walks in the weighted extended Cayley networks, focusing on a specific case with a perfect trap located at the central node. Then, the scaling expression of the average trapping time (ATT) is derived based on the layering of weighted extended Cayley networks. It was surprisingly found that (1) Regardless of the relationship between m and n, the dominant terms of ATTs are consistent. (2) ATTs are less affected by the structural parameter m and the weight factor r when r is less than or equal to the ratio of n to m−1, indicating that the efficiency of the trapping process is independent of m and r. (3) When r is greater than the ratio of n to m−1, the efficiency of the trapping process depends on three main parameters: two structural parameters of the networks m, n and a weight factor r, which means that the smaller the multiplier of three numbers r, n and m − 1 is, the more efficient the trapping process is. Therefore, the trapping efficiency of the weighted extended Cayley networks is not only affected by the underlying structures of the networks m and n, but also by the weight factor r.

VL  - 12
IS  - 5
ER  -

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