  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.
