Coherence Scaling of Noisy Second-Order Scale-Free Consensus Networks

A striking discovery in the field of network science is that the majority of real networked systems have some universal structural properties. In general, they are simultaneously sparse, scale-free, small-world, and loopy. In this article, we investigate the second-order consensus of dynamic networks with such universal structures subject to white noise at vertices. We focus on the network coherence H_SO characterized in terms of the H_₂-norm of the vertex systems, which measures the mean deviation of vertex states from their average value. We first study numerically the coherence of some representative real-world networks. We find that their coherence H_SO scales sublinearly with the vertex number N. We then study analytically H_SO for a class of iteratively growing networks--pseudofractal scale-free webs (PSFWs), and obtain an exact solution to HSO, which also increases sublinearly in N, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study H_SO for Sierpinśki gaskets, for which H_SO  grows superlinearly in N, with a power exponent much larger than 1. Sierpinśki gaskets have the same number of vertices and edges as the PSFWs but do not display the scale-free and small-world properties. We thus conclude that the scale-free, small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of H_SO.