Consensus dynamics and coherence in hierarchical small-world networks

TL;DR

This paper analyzes the consensus dynamics in hierarchical small-world networks, revealing their fast convergence, limited delay robustness, and optimal structure for noisy consensus, using spectral graph theory.

Contribution

It explicitly determines Laplacian eigenvalues for hierarchical small-world networks and analyzes their consensus properties, answering open questions and linking network structure to robustness.

Findings

Faster convergence than some sparse networks.

Less robust to time delays.

Optimal structure for noisy consensus dynamics.

Abstract

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.