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：层次小世界网络是一种真实网络，它很好地模拟了中国及许多其他国家传销中的利益传递网络。本文运用谱图理论，研究了层次小世界网络中共识问题的三个重要方面：收敛速度、通信时延鲁棒性及网络一致性。首先，利用其树状结构显式确定了层次小世界网络的拉普拉斯特征值。其次，我们发现层次小世界网络上的共识算法比某些已充分研究的稀疏网络收敛更快，但对时延的鲁棒性较弱。此外，本文还推导了一阶和二阶网络一致性的闭合解析式。结果表明，层次小世界网络在噪声共识动态中具有最优结构。因此，我们为Yi等人提出的两个开放问题提供了肯定性回答。最后，我们论证了网络结构特征（如大最大度数、短平均路径长度、高顶点连通性和边连通性）是强鲁棒性（针对外部扰动）的原因。