Coherence Scaling of Noisy Second-Order Scale-Free Consensus Networks

Abstract

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 $H_{\mathrm{SO}}$ 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 $H_{\mathrm{SO}}$ scales sublinearly with the vertex number $N$. We then study analytically $H_{\mathrm{SO}}$ for a class of iteratively growing networks -- pseudofractal scale-free webs (PSFWs), and obtain an exact solution to $H_{\mathrm{SO}}$, which also increases sublinearly in $N$, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study $H_{\mathrm{SO}}$ for Sierpinśki gaskets, for which $H_{\mathrm{SO}}$ 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 and small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of $H_{\mathrm{SO}}$.

References in corpus (2)

• Domination number and minimum dominating sets in pseudofractal scale-free web and Sierpiński graph
• Biharmonic Distance and the Performance of Second-Order Consensus Networks with Stochastic Disturbances

Cited by in corpus (2)

• Optimal Scale-Free Small-World Graphs with Minimum Scaling of Cover Time
• Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web