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 <italic>H</italic>_SO characterized in terms of the <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}_{2}$ </tex-math></inline-formula>-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 <italic>H</italic>_SO scales sublinearly with the vertex number <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>. We then study analytically <italic>H</italic>_SO for a class of iteratively growing networks—pseudofractal scale-free webs (PSFWs), and obtain an exact solution to <italic>H</italic>_SO, which also increases sublinearly in <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study <italic>H</italic>_SO for Sierpinśki gaskets, for which <italic>H</italic>_SO grows superlinearly in <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>, 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 <italic>H</italic>_SO.