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：层级小世界网络是一种现实世界网络，能够很好地模拟中国及许多其他国家传销活动中的利益传递网络。本文利用谱图理论，研究了层级小世界网络中共识问题的三个重要方面：收敛速度、通信时延鲁棒性以及网络相干性。首先，我们利用其树状结构显式确定了层级小世界网络的拉普拉斯特征值。其次，我们发现层级小世界网络上的共识算法比某些经过充分研究的稀疏网络上的算法收敛更快，但对时延的鲁棒性较差。此外，我们还推导出了一阶和二阶网络相干性的闭式解。结果表明，层级小世界网络具有噪声共识动力学的最优结构。因此，我们为Yi等人提出的两个开放性问题给出了肯定答案。最后，我们认为网络的大最大度、小平均路径长度以及大顶点和边连通性等结构特征，是导致其对外部扰动具有强鲁棒性的原因。