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 H₂ -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 H_SO, 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.