The hierarchical small-world network is a real-world network. It models well the benefit transmission web of the pyramid selling in China and many other countries. In this paper, by applying the spectral graph theory, we study three important aspects of the consensus problem in the hierarchical small-world network: convergence speed, communication time-delay robustness, and network coherence. Firstly, we explicitly determine the Laplacian eigenvalues of the hierarchical small-world network by making use of its treelike structure. Secondly, we find that the consensus algorithm on the hierarchical small-world network converges faster than that on some well-studied sparse networks, but is less robust to time delay. The closed-form of the first-order and the second-order network coherence are also derived. Our result shows that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Therefore, we provide a positive answer to two open questions of Yi \emph{et al}. Finally, we argue that some network structure characteristics, such as large maximum degree, small average path length, and large vertex and edge connectivity, are responsible for the strong robustness with respect to external perturbations.

翻译：上层小世界网络是一个真实的世界网络。 它在中国和许多其他国家模拟了金字塔销售的收益传输网。 在本文中, 通过应用光谱图理论, 我们研究了上层小世界网络的共识问题的三个重要方面: 趋同速度、 通信时间间隔强性和网络一致性。 首先, 我们使用像树一样的结构, 明确确定上层小世界网络的Lapalcian eigen值。 其次, 我们发现, 上层小世界网络的协商一致算法比一些研究良好的稀疏网络的趋近速度快, 但却不太耐久。 一阶和二阶网络一致性的封闭形式也被推导出。 我们的结果显示, 上层小世界网络有一个最优的杂乱的共识动态结构。 因此, 我们给出了两个开放的小世界网络结构问题的正面答案。 最后, 我们认为, 一些网络结构特征, 如大的最大程度、 小平均路径长度, 以及大左端和边缘连接性, 对外部的坚固性负有责任 。