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 generally, they are simultaneously sparse, scale-free, small-world, and loopy. In this paper, 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 $H_{\rm SO}$ characterized in terms of the $\mathcal{H}_2$-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 $H_{\rm SO}$ scales sublinearly with the vertex number $N$. We then study analytically $H_{\rm SO}$ for a class of iteratively growing networks -- pseudofractal scale-free webs (PSFWs), and obtain an exact solution to $H_{\rm SO}$, which also increases sublinearly in $N$, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study $H_{\rm SO}$ for Sierpin\'ski gaskets, for which $H_{\rm SO}$ grows superlinearly in $N$, with a power exponent much larger than 1. Sierpin\'ski 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 and small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of $H_{\rm SO}$.