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
$\mathcal {H}_{2}$
-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
_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
.