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Scaling of average weighted shortest path and average receiving time on weighted hierarchical networks

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  • Sun, Yu
  • Dai, Meifeng
  • Xi, Lifeng

Abstract

Recent work on the networks has focused on the weighted hierarchical networks that are significantly different from the un-weighted hierarchical networks. In this paper we study a family of weighted hierarchical networks which are recursively defined from an initial uncompleted graph, in which the weights of edges have been assigned to different values with certain scale. Firstly, we study analytically the average weighted shortest path (AWSP) on the weighted hierarchical networks. Using a recursive method, we determine explicitly the AWSP. The obtained rigorous solution shows that the networks grow unbounded but with the logarithm of the network size, while the weighted shortest paths stay bounded. Then, depending on a biased random walk, we research the mean first-passage time (MFPT) between a hub node and any peripheral node. Finally, we deduce the analytical expression of the average of MFPTs for a random walker originating from any node to first visit to a hub node, which is named as the average receiving time (ART). The obtained result shows that ART is bounded or grows sublinearly with the network order relating to the number of initial nodes and the weighted factor or grows quadratically with the iteration.

Suggested Citation

  • Sun, Yu & Dai, Meifeng & Xi, Lifeng, 2014. "Scaling of average weighted shortest path and average receiving time on weighted hierarchical networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 407(C), pages 110-118.
    • Handle: RePEc:eee:phsmap:v:407:y:2014:i:c:p:110-118
    DOI: 10.1016/j.physa.2014.03.088
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    References listed on IDEAS

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    1. Fagiolo, Giorgio & Reyes, Javier & Schiavo, Stefano, 2008. "On the topological properties of the world trade web: A weighted network analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3868-3873.
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    Citations

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    Cited by:

    1. Wijesundera, Isuri & Halgamuge, Malka N. & Nirmalathas, Ampalavanapillai & Nanayakkara, Thrishantha, 2016. "MFPT calculation for random walks in inhomogeneous networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 986-1002.
    2. Ye, Dandan & Dai, Meifeng & Sun, Yu & Su, Weiyi, 2017. "Average weighted receiving time on the non-homogeneous double-weighted fractal networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 390-402.
    3. Zong, Yue & Dai, Meifeng & Wang, Xiaoqian & He, Jiaojiao & Zou, Jiahui & Su, Weiyi, 2018. "Network coherence and eigentime identity on a family of weighted fractal networks," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 184-194.
    4. Dai, Meifeng & Shao, Shuxiang & Su, Weiyi & Xi, Lifeng & Sun, Yanqiu, 2017. "The modified box dimension and average weighted receiving time of the weighted hierarchical graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 475(C), pages 46-58.
    5. Niu, Min & Song, Shuaishuai, 2018. "Scaling of average weighted shortest path and average receiving time on the weighted Cayley networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 707-717.
    6. Jiao, Bo & Nie, Yuan-ping & Shi, Jian-mai & Huang, Cheng-dong & Zhou, Ying & Du, Jing & Guo, Rong-hua & Tao, Ye-rong, 2016. "Scaling of weighted spectral distribution in deterministic scale-free networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 451(C), pages 632-645.

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