Full Text:   <484>

Summary:  <45>

CLC number: O155; TP11

On-line Access: 2020-06-12

Received: 2019-03-08

Revision Accepted: 2019-06-23

Crosschecked: 2019-08-09

Cited: 0

Clicked: 765

Citations:  Bibtex RefMan EndNote GB/T7714

 ORCID:

Wei-gang Sun

http://orcid.org/0000-0001-8699-5392

-   Go to

Article info.
Open peer comments

Frontiers of Information Technology & Electronic Engineering  2020 Vol.21 No.6 P.931-938

http://doi.org/10.1631/FITEE.1900133


Coherence analysis and Laplacian energy of recursive trees with controlled initial states


Author(s):  Mei-du Hong, Wei-gang Sun, Su-yu Liu, Teng-fei Xuan

Affiliation(s):  School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China

Corresponding email(s):   wgsun@hdu.edu.cn

Key Words:  Consensus, Network coherence, Laplacian energy


Mei-du Hong, Wei-gang Sun, Su-yu Liu, Teng-fei Xuan. Coherence analysis and Laplacian energy of recursive trees with controlled initial states[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(6): 931-938.

@article{title="Coherence analysis and Laplacian energy of recursive trees with controlled initial states",
author="Mei-du Hong, Wei-gang Sun, Su-yu Liu, Teng-fei Xuan",
journal="Frontiers of Information Technology & Electronic Engineering",
volume="21",
number="6",
pages="931-938",
year="2020",
publisher="Zhejiang University Press & Springer",
doi="10.1631/FITEE.1900133"
}

%0 Journal Article
%T Coherence analysis and Laplacian energy of recursive trees with controlled initial states
%A Mei-du Hong
%A Wei-gang Sun
%A Su-yu Liu
%A Teng-fei Xuan
%J Frontiers of Information Technology & Electronic Engineering
%V 21
%N 6
%P 931-938
%@ 2095-9184
%D 2020
%I Zhejiang University Press & Springer
%DOI 10.1631/FITEE.1900133

TY - JOUR
T1 - Coherence analysis and Laplacian energy of recursive trees with controlled initial states
A1 - Mei-du Hong
A1 - Wei-gang Sun
A1 - Su-yu Liu
A1 - Teng-fei Xuan
J0 - Frontiers of Information Technology & Electronic Engineering
VL - 21
IS - 6
SP - 931
EP - 938
%@ 2095-9184
Y1 - 2020
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/FITEE.1900133


Abstract: 
We study the consensus of a family of recursive trees with novel features that include the initial states controlled by a parameter. The consensus problem in a linear system with additive noises is characterized as network coherence, which is defined by a Laplacian spectrum. Based on the structures of our recursive treelike model, we obtain the recursive relationships for Laplacian eigenvalues in two successive steps and further derive the exact solutions of first- and second-order coherences, which are calculated by the sum and square sum of the reciprocal of all nonzero Laplacian eigenvalues. For a large network size N, the scalings of the first- and second-order coherences are lnN and N$, respectively. The smaller the number of initial nodes, the better the consensus bears. Finally, we numerically investigate the relationship between network coherence and laplacian energy, showing that the first- and second-order coherences increase with the increase of laplacian energy at approximately exponential and linear rates, respectively.

具有受控初始状态递归树的一致性分析及其拉普拉斯能量

洪美都,孙伟刚,刘苏雨,轩腾飞
杭州电子科技大学理学院,中国杭州市,310018

摘要:本文研究一类具有受控初始状态递归树的一致性问题。由拉普拉斯谱定义的网络一致性用于刻画含有噪声线性系统的一致性动力学。基于这类递归树的规则结构,得到拉普拉斯特征值连续两次迭代的递归关系,并由此得到一阶和二阶一致性的精确解。它们由所有非零拉普拉斯特征值的倒数和与平方和来定义。一阶和二阶一致性的幂律关于网络规模N分别为lnNNN。研究表明递归树初始节点数目越少,其一致性表现越好。最后,用数值例子研究一致性和拉普拉斯能量之间的关系,结果表明一阶和二阶一致性分别随拉普拉斯能量以指数和线性速率增长。

关键词:一致性;网络一致性;拉普拉斯能量

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1]Albert R, Barabási AL, 2002. Statistical mechanics of complex networks. Rev Mod Phys, 74(1):47-97.

[2]Bamieh B, Jovanovic MR, Mitra P, et al., 2012. Coherence in large-scale networks: dimension-dependent limitations of local feedback. IEEE Trans Autom Contr, 57(9): 2235-2249.

[3]Chu ZQ, Liu JB, Li XX, 2016. The Laplacian-energy-like invariants of three types of lattices. J Anal Methods Chem, 2016:7320107.

[4]Dai MF, He JJ, Zong Y, et al., 2018. Coherence analysis of a class of weighted networks. Chaos, 28(4):043110.

[5]Dorogovtsev SN, Mendes JFF, 2002. Evolution of networks. Adv Phys, 51(4):1079-1187.

[6]Dorogovtsev SN, Goltsev AV, Mendes JFF, et al., 2003. Spectra of complex networks. Phys Rev E, 68(4):046109.

[7]Farkas IJ, Derényi I, Barabási AL, et al., 2001. Spectra of “real-world” graphs: beyond the semicircle law. Phys Rev E, 64(2):026704.

[8]Goh KI, Kahng B, Kim D, 2001. Spectra and eigenvectors of scale-free networks. Phys Rev E, 64(5):051903.

[9]Gutman I, Zhou B, 2006. Laplacian energy of a graph. Linear Algebra Appl, 414(1):29-37.

[10]Ma CQ, Li T, Zhang JF, 2010. Consensus control for leader-following multi-agent systems with measurement noises. J Syst Sci Complex, 23(1):35-49.

[11]Newman ME, 2003. The structure and function of complex networks. SIAM Rev, 45(2):167-256.

[12]Patterson S, Bamieh B, 2014. Consensus and coherence in fractal networks. IEEE Trans Contr Netw Syst, 1(4): 338-348.

[13]Robbiano M, Jimenez R, 2009. Applications of a theorem by Ky fan in the theory of Laplacian energy of graphs. Match-Commun Math Comput Chem, 62(3):537-552.

[14]Russo G, Shorten R, 2018. On common noise-induced synchronization in complex networks with state-dependent noise diffusion processes. Phys D, 369:47-54.

[15]Song L, Huang D, Nguang SK, et al., 2016. Mean square consensus of multi-agent systems with multiplicative noises and time delays under directed fixed topologies. Int J Contr Autom Syst, 14(1):69-77.

[16]Strogatz SH, 2001. Exploring complex networks. Nature, 410(6825):268-276.

[17]Sun WG, Ding QY, Zhang JY, et al., 2014. Coherence in a family of tree networks with an application of Laplacian spectrum. Chaos, 24(4):043112.

[18]Wang LS, Zhang JB, Sun WG, 2018. Adaptive outer synchronization and topology identification between two complex dynamical networks with time-varying delay and disturbance. IMA J Math Contr Inform, in press.

[19]Xiao L, Boyd S, Kim SJ, 2007. Distributed average consensus with least-mean-square deviation. J Parall Distrib Comput, 67(1):33-46.

[20]Yi YH, Zhang ZZ, Lin Y, et al., 2015. Small-world topology can significantly improve the performance of noisy consensus in a complex network. Comput J, 58(12):3242-3254.

[21]Zhang ZZ, Qi Y, Zhou SG, et al., 2009. Exact solution for mean first-passage time on a pseudofractal scale-free web. Phys Rev E, 79(2):021127.

[22]Zong Y, Dai MF, Wang XQ, et al., 2018. Network coherence and eigentime identity on a family of weighted fractal networks. Chaos Sol Fract, 109:184-194.

Open peer comments: Debate/Discuss/Question/Opinion

<1>

Please provide your name, email address and a comment





Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - Journal of Zhejiang University-SCIENCE