Name
Abstract | ||
|---|---|---|
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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</sub>
characterized in terms of the
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</sub>
scales sublinearly with the vertex number
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>
. We then study analytically
<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</sub>
for a class of iteratively growing networks—pseudofractal scale-free webs (PSFWs), and obtain an exact solution to
<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</sub>
, which also increases sublinearly in
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</sub>
for Sierpinśki gaskets, for which
<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</sub>
grows superlinearly in
<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</sub>
. |
Year | DOI | Venue |
|---|---|---|
2022 | 10.1109/TCYB.2021.3052519 | IEEE Transactions on Cybernetics |
Keywords | DocType | Volume |
Distributed average consensus,Gaussian white noise,multiagent systems,network coherence,scale-free network,small-world network | Journal | 52 |
Issue | ISSN | Citations |
7 | 2168-2267 | 0 |
PageRank | References | Authors |
0.34 | 36 | 6 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Wanyue Xu | 1 | 1 | 3.07 |
| Bin Wu | 2 | 53 | 2.88 |
| Zuobai Zhang | 3 | 1 | 2.39 |
| Zhongzhi Zhang | 4 | 85 | 22.02 |
| Haibin Kan | 5 | 0 | 0.34 |
| Guanrong Chen | 6 | 2 | 1.04 |