Consensus dynamics on a family of weighted recursive trees

Abstract

The consensus dynamics with additive stochastic disturbances are characterized by the network coherence, which is the robustness of consensus algorithms when the nodes are subject to external perturbations. In this paper, the research goal is to obtain the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. One innovation point of this paper is the structure of a family of the weighted recursive trees with weight factor. We mainly obtain the exact expressions and scalings of network coherence on the family of weighted recursive trees. The scalings of first-order network coherence with network size obey three laws along with the range of the weight factor, while the scalings of second-order network coherence obey four laws along with the range of the weight factor. In addition, the scalings of first- and second-order network coherence on our studied networks are smaller than those performed on other studied networks when [Formula: see text]. The obtained results indicate that the efficiency of network coherence on the weighted network has close relation to the weight distribution, and we can design a better weight distribution to make the coherence of network more efficient.