In this paper, we propose a family of nested weighted n-polygon networks, which is a kind of promotion of infinite fractal dimension networks. We study the coherence of the networks with recursive features that contain the initial states dominated by a weighted parameter. The network coherence is a consensus problem with additive noises, and it is known that coherence is defined by the eigenvalues of the Laplacian matrix. According to the structure of recursive nested model, we get the recursive expressions of Laplacian eigenvalues and further derive the exact results of first- and second-order coherence. Finally, we investigate the influential impacts on the coherence for different large parameters and discuss the relationship between Laplacian energy and network coherence. Furthermore, we obtain the expressions of Kirchhoff index, mean first-passage time and average path length of the networks.

中文翻译：

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嵌套加权n-多边形网络族的网络一致性分析

在本文中，我们提出了一系列嵌套加权n-多边形网络，是无限分形维数网络的一种推广。我们研究具有递归特征的网络的连贯性，这些特征包含由加权参数控制的初始状态。网络相干性是一个带有加性噪声的共识问题，众所周知，相干性是由拉普拉斯矩阵的特征值定义的。根据递归嵌套模型的结构，得到拉普拉斯特征值的递归表达式，进一步推导出一阶和二阶相干的精确结果。最后，我们研究了不同大参数对相干性的影响，并讨论了拉普拉斯能量与网络相干性之间的关系。此外，我们得到了网络的基尔霍夫指数、平均首次通过时间和平均路径长度的表达式。