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 HSO 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 HSO scales sublinearly with the vertex number N . We then study analytically HSO for a class of iteratively growing networks—pseudofractal scalefree 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 HSO for Sierpinśki gaskets, for which HSO 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 and small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of HSO.