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

Mei-du HONG; Wei-gang SUN; Su-yu LIU; Teng-fei XUAN

doi:10.1631/FITEE.1900133

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

Regular Papers | Updated：2022-05-19

- Coherence analysis and Laplacian energy of recursive trees with controlled initial states
- 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
- Frontiers of Information Technology & Electronic Engineering Vol. 21, Issue 6, Pages: 931-938(2020)
- Affiliations：
  
  School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
- Author bio：
  
  Wei-gang SUN, E-mail: wgsun@hdu.edu.cn
- Funds：
- DOI：10.1631/FITEE.1900133
  CLC： O155; TP11
- Received：08 March 2019，
  
  Revised：2019-;8-;9，
  
  Published Online：05 September 2019，
  
  Published：2020-06
- Accepted：
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Mei-du HONG, Wei-gang SUN, Su-yu LIU, et al. Coherence analysis and Laplacian energy of recursive trees with controlled initial states[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(6): 931-938.
DOI：

Mei-du HONG, Wei-gang SUN, Su-yu LIU, et al. Coherence analysis and Laplacian energy of recursive trees with controlled initial states[J]. Frontiers of Information Technology & Electronic Engineering, 2020, 21(6): 931-938. DOI： 10.1631/FITEE.1900133.

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

分别为ln

和

N。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。

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

the scalings of the first- and second-order coherences are ln

and

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.

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Keywords

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