Research Article | | Peer-Reviewed

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

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), 2023. Published by Science Publishing Group

Keywords

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

References
[1] Meifeng Dai and Danping Zhang. A weighted evolving network with aging-node-deleting and local rearrangements of weights. International Journal of Modern Physics C, 2014, 25 (02): 47-.
[2] Lei. H, Li. T, Ma. YD, Wang. H. Analyzing lattice networks through substructures [J]. Applied Mathematics and Computation, 2018, 329: 297-314.
[3] Lucas S. Flores; Marco A. Amaral; Mendeli H. Vainstein; Heitor C. M. Fernandes. Cooperation in regular lattices [J]. Chaos, Solitons & Fractals, 2022£o112744.
[4] Robert M, Ziff. Explosive Growth in Biased Dynamic Percolation on Two-Dimensional Regular Lattice Networks [J]. Physical Review Letters 2009, 103 (04): 045701 (1-4).
[5] Garza-López, Roberto A; Linares, Anthony; Yoo, Alice; Evans, Greg; Kozak, John J. Invariance relations for random walks on simple cubic lattices [J]. Chemical Physics Letters, 2006, 421 (NO. 1-3): 287-294.
[6] Garza-López, Roberto A; Kozak, John J. Invariance relations for random walks on square-planar lattices [J]. Chemical Physics Letters, 2005, 406 (NO. 1-3): 38-43.
[7] Garza-López, Roberto A; Kozak, John J. Invariance relations for random walks on hexagonal lattices [J]. Chemical Physics Letters, 2003, 371 (03): 365-370.
[8] John J. Kozak and V. Balakrishnan. Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket [J]. Physical Review. E, 2002, 65 (02): 021105.
[9] V. Balakrishnan and John J. Kozak. Analytic expression for the mean time to absorption for a random walker on the Sierpinski fractal. III. The effect of non-nearest- neighbor jumps [J]. PHYSICAL REVIEW E, 2013, 88 (05): 52139-52139.
[10] Jonathan L Bentz; John W Turner; John J Kozak. Analytic expression for the mean time to absorption for a random walker on the Sierpinski gasket. II. The eigenvalue spectrum [J]. Physical review. E, Statistical, nonlinear, and soft matter physics, 2010, 82 (01-01): 011137.
[11] John J. Kozak and V. Balakrishnan. Exact Formula for the Mean Length of a Random Walk on the Sierpinski Tower [J]. International Journal of Bifurcation and Chaos, 2002, 12 (11): 2379-2385.
[12] Zhang, ZZ; Zhou, SG; Xie, WL; Chen, LC; Lin, Y; Guan, JH. Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect [J]. Physical review. E, Statistical, nonlinear, and soft matter physics, 2009, 79 (06-01): 061113.
[13] Zhang, JY; Sun, WG; Chen, GR. Exact scaling for the mean first-passage time of random walks on a generalized Koch network with a trap [J].Chinese Physics B, 2012, 21 (03): 038901.
[14] Su, Jing; Zhang, Mingjun; Yao, Bing. The Structure and First-Passage Properties of Generalized Weighted Koch Networks [J]. Entropy, 2022, 24 (03): 409.
[15] Zhang, JY and Sun, WG. The structural properties of the generalized Koch network [J]. Journal of Statistical Mechanics: Theory and Experiment, 2010, 2010 (07): P07011.
[16] E. Agliari. Exact mean first-passage time on the T-graph [J]. Physical review, E. Statistical, nonlinear, and soft matter physics, 2008, 77 (01-01): 011128 (1-6).
[17] Lin, Yuan; Zhang, Zhongzhi; Wu, Bin. Determining mean first-passage time on a class of treelike regular fractals [J]. Physical Review E, 2010, 82 (03): 031140.
[18] Zhang, Zhongzhi; Wu, Bin; Chen, Guanrong. Complete spectrum of the stochastic master equation for random walks on treelike fractals [J]. Europhysics Letters, 2011, 96 (4): 2510-2513.
[19] Julaiti, Alafate; Wu, Bin; Zhang, Zhongzhi. Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications [J]. Journal of Chemical Physics, 2013, 138 (20): 204116.
[20] Peng J and Xu G. Efficiency analysis of diffusion on T- fractals in the sense of random walks [J]. The Journal of Chemical Physics, 2014, 140 (13): 134102.
[21] Roberts AP and Haynes CP. Global first-passage times of fractal lattices [J]. Physical review, E, 2008, 78 (04-01): 041111 (1-9).
[22] S. N Dorogovtsev, A. V Goltsev, J. F. F Mendes. Pseudofractal scale-free web [J]. PHYSICAL REVIEW E, 2002, 65 (6): 066122.
[23] Zhang, ZZ; Qi, Y; Zhou, SG; Xie, WL; Guan, JH. Exact solution for mean first-passage time on a pseudofractal scale-free web [J]. Physical review, E, 2009, 79 (02-01): 021127 (1-6).
[24] Peng, JH; Agliari, E; Zhang, ZZ. Exact calculations of first-passage properties on the pseudofractal scale-free web [J]. CHAOS, 2015, 25 (07): 073118.
[25] Wu, B. The average trapping time on the weighted pseudofractal scale-free web [J]. Journal of Statistical Mechanics: Theory and Experiment, 2020, 2020 (04): 043209.
[26] Zhang, ZZ; Chen, LC; Zhou, SG; Fang, L; Guan, JH; Zou, T. Analytical solution of average path length for Apollonian networks [J]. Physical review, E, 2008, 77 (01-02): 017102 (1-4).
[27] José S Andrade; Hans J Herrmann; Roberto F S Andrade; Luciano R da Silva. Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs [J]. Physical review letters, 2005, 94 (01): 018702.
[28] Zhang, Zhongzhi; Guan, Jihong; Xie, Wenlei; Qi, Yi; Zhou, Shuigeng. Random walks on the Apollonian network with a single trap [J]. EPL, 2009, 86 (01): 10006.
[29] Zhang, ZZ; Rong, LL; Zhou, SG. Evolving Apollonian networks with small-world scale-free topologies [J]. Physical review, E, 2006, 74 (04-02): 6105 (1-9).
[30] Zhang, ZZ; Lin, YA; Gao, SY; Zhou, SG; Guan, JH; Li, M. Trapping in scale-free networks with hierarchical organization of modularity [J]. Physical review, E, 2009, 80 (05-01): 051120 (1-10).
[31] Yang, Y; Lin, Y; Zhang, Z. Random walks in modular scale-free networks with multiple traps [J]. Physical review, E, 2012, 85 (01-01): 011106 (1-10).
[32] Niu, Min; Shao, Mengjun. The asymptotic formula on average weighted path length for scale-free modular network [J]. Modern Physics Letters B, 2021, 35 (18): 2150298.
[33] Bentz J L, Hosseini F N, Kozak J J. Influence of geometry on light harvesting in dendrimeric systems [J]. Chemical Physics Letters, 2003, 370 (3-4); 319-326
[34] Li L, Guan J, Zhou S. Mean first passage time for random walk on dual structure of dendrimer [J]. Physica A Statistical Mechanics & Its Applications, 2014, 415: 463-472.
[35] Argyrakis P, Kopelman R. Random walks and reactions on dendrimer structures [J]. Chemical Physics, 2000, 261 (3): 391-398.
[36] Ma, Fei; Wang, Xiaomin; Wang, Ping; Luo, Xudong. Random walks on the generalized Vicsek fractal [J]. EPL, 2021, 133 (04): 40004.
[37] Dolgushev, Maxim; Liu, Hongxiao; Zhang, Zhongzhi. Extended Vicsek fractals: laplacian spectra and their applications [J]. Physical Review E, 2016, 94 (05): 052501.
[38] A. Blumen; Ch. von Ferber; A. Jurjiu; Th. Koslowski. Generalized Vicsek Fractals: Regular Hyperbranched Polymers [J]. Macromolecules, 2004, 37 (02): 638-650.
[39] Zhongzhi Zhang; Shuigeng Zhou; Lichao Chen; Ming Yin; Jihong Guan. Exact solution of mean geodesic distance for Vicsek fractals [J]. Journal of physics, A. Mathematical and theoretical, 2008, 41 (48): 485102.
[40] Zhang, ZZ; Wu, B; Zhang, HJ; Zhou, SG; Guan, JH; Wang, ZG. Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices [J]. Physical Review E, 2010, 81 (03- 01): 031118.
[41] D. Y Yan, C. Gao, and H. Frey. Hyperbranched Polymers [J]. Chemical Society Reviews, 2011: 251-271.
[42] Xu, Ming; Xu, Chuanyun; Wang, Huan; Deng, Congzheng; Cao, KeiFei. Analytical controllability of deterministic scale-free networks and Cayley trees [J]. European Physical Journal B, 2015, 88 (07): 168.
[43] Lin Y, Zhang Z. Influence of trap location on the efficiency of trapping in dendrimers and regular hyperbranched polymers [J]. Journal of Chemical Physics, 2013, 138 (9): 088701-R.
[44] Wu B, Lin Y, Zhang ZZ, Chen, GR. Trapping in dendrimers and regular hyperbranched polymers [J]. Journal of Chemical Physics, 2012, Vol. 137 (04): 044903.
[45] Marc Barthélemy; Alain Barrat; Romualdo Pastor- Satorras; Alessandro Vespignani. Characterization and modeling of weighted networks [J]. Physica A: Statistical Mechanics and its Applications, 2005, 346 (1-2): 34-43.
[46] Carletti T and Righi S. Weighted Fractal Networks [J]. Physica A: Statal Mechanics and its Applications, 2009, 389 (10): 2134-2142.
[47] Dai, MF; Zong, Y; He, JJ; Sun, Y; Shen, CY; Su, WY. The trapping problem of the weighted scale-free treelike networks for two kinds of biased walks [J]. Chaos, 2018, 28 (11): 113115.
[48] Dai, MF; Liu, J; Li, XY. Trapping time of weighted- dependent walks depending on the weight factor [J]. Chaos Solitons & Fractals, 2014, 60: 49-55.
[49] Ann E Krause; Kenneth A Frank; Doran M Mason; Robert E Ulanowicz; William W Taylor. Compartments revealed in food-web structure [J]. Nature, 2003, 426 (6964): 282.
[50] Chen J, Dai MF, Wen ZX, Xi LF. Trapping on modular scale-free and small-world networks with multiple hubs [J]. Physica A Statistical Mechanics & Its Applications, 2014, 393: 542-552.
[51] Shlomo Havlin; Daniel Ben-Avraham. Diffusion in disordered media [J]. Advances in physics, 2002, 51 (01): 187-292.
[52] Lloyd, Alun L. How viruses spread among computers and people [J]. Science, 2001, 292 (5520): 1316-1317.
[53] Shlesinger MF. Mathematical physics: search research [J]. Nature, 2006, 443 (7109): 281.
[54] Dandan Ye, Meifeng Dai, Yanqiu Sun, Shuxiang Shao, Qi Xie. Average receiving scaling of the weighted polygon Koch networks with the weight-dependent walk [J]. Physica A - Statistical mechanics and its applications, 2016, 458: 1-8.
[55] Stepanenko IA, Kompanets VO, Chekalin SV, et al. Photosynthetic light-harvesting complexes: fluorescent and absorption spectroscopy under two-photon (1200- 1500 nm) and one-photon (600-750 nm) excitation by laser femtosecond pulses [J]. proc spie, 2011, 7994 (2): 170-187.
[56] Bentz JL, Kozak JJ. Influence of geometry on light harvesting in dendrimeric systems. II. nth-nearest neighbor effects and the onset of percolation [J]. Journal of Luminescence, 2006, 121 (1): 62-74.
[57] Agliari E. Trapping of Continuous-Time Quantum walks on Erdos-Renyi graphs [J]. Physica A Statistical Mechanics & Its Applications, 2011, 390 (11): 1853- 1860.
[58] Sokolov IM, Mai J, Blumen A. Paradoxal Diffusion in Chemical Space for Nearest-Neighbor Walks over Polymer Chains [J]. Physical Review Letters, 1997, 79 (5): 857-860.
[59] Blumen, A. Energy transfer as a random walk on regular lattices [J]. Journal of Chemical Physics, 1981, 75 (2): 892-907.
[60] Dai, Meifeng; Dai, Changxi; Wu, Huiling; Wu, Xianbin; Feng, Wenjing; Su, Weiyi. The trapping problem and the average shortest weighted path of the weighted pseudofractal scale-free networks [J]. INTERNATIONAL JOURNAL OF MODERN PHYSICS C, 2019, 30 (01): 1950010.
[61] Dai, Meifeng; Ju, Tingting; Zong, Yue; Fie, Jiaojiao; Shen, Chunyu; Su, Weiyi. TRAPPING PROBLEM OF THE WEIGHTED SCALE-FREE TRIANGULATION NETWORKS FOR BIASED WALKS [J]. FRACTALS- COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY, 2019, 27 (03).
[62] Dai, Meifeng; Xie, Qi; Xi, Lifeng. TRAPPING ON WEIGHTED TETRAHEDRON KOCH NETWORKS WITH SMALL-WORLD PROPERTY [J]. FRACTALS- COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY, 2014, 22 (1-2): 1-9.
[63] Zhang ZZ, Li H, Yi Y. Anomalous behavior of trapping in extended dendrimers with a perfect trap [J]. The Journal of Chemical Physics, 2015, 143 (6): 064901.
[64] Niu M, Song SS. Scaling of average weighted shortest path and average receiving time on the weighted Cayley networks [J]. Physica A: Statal Mechanics and its Applications, 2018, 506 (01): 707-717.
[65] S. Boccaletti, et al. Complex networks: Structure and dynamics [J]. Physics Reports, 2006, 424 (4-5): 175-308.
[66] Sun Y, Dai MF, Xi LF. Scaling of average weighted shortest path and average receiving time on weighted hierarchical networks [J]. Physica, A. Statistical mechanics and its applications, 2014, 407: 110-118.
[67] Dai MF, Chen DD, Dong YJ, Liu JE. Scaling of average receiving time and average weighted shortest path on weighted Koch networks [J]. Physica A - Statistical mechanics and its applications, 2012, 391 (23): 6165- 6173.
[68] Dandan Ye, Song Liu, Jia Li, Fei Zhang, Changling Han, Wei Chen, Yingze Zhang. Scaling of average receiving time on weighted polymer networks with some topological properties [J]. Scientific Reports. 2017, 07 (01): 2128.
[69] Fei Zhang, Dandan Ye, Changling Han, Wei Chen, Yingze Zhang. The average weighted receiving time with weight-dependent walk on a family of double- weighted polymer networks [J]. International Journal of Modern Physics C: Computational Physics & Physical Computation. 2019, 30 (08): 1950063.
[70] Meifeng Dai, Dandan Ye, Jie Hou, Lifeng Xi, Weiyi Su. Average weighted trapping time of the node- and edge-weighted fractal networks [J]. Communications in nonlinear science and numerical simulation, 2016, 39 (10): 209-219.
[71] Dandan Ye, Meifeng Dai, Yu Sun, Weiyi Su. Average weighted receiving time on the non-homogeneous double-weighted fractal networks [J]. Physica A - Statistical mechanics and its applications, 2017, 473: 390-402.
[72] Condamin S, Bénichou O, Tejedor V, Voituriez R, Klafter J. First-passage times in complex scale-invariant media [J]. Nature 2007; 450 (7166): 77-80.
[73] Meifeng Dai, Dandan Ye, Jie Hou, Lifeng Xi. Scaling of average weighted receiving time on double-weighted koch networks [J]. Fractals-Complex Geometry Patterns and Scaling in Nature and Society, 2015, 23 (02): 1550011.
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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
  • Institute of Orthopaedics, the Third Hospital of Hebei Medical University, Shijiazhuang, China

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

  • Hebei Medical University Basical Medicine College, Shijiazhuang, China

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

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

  • Hebei Orthopedic Clinical Research Center, Shijiazhuang, China

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

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

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