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Edited by: Jia-Bao Liu, Anhui Jianzhu University, China

Reviewed by: Junhao Peng, Guangzhou University, China; Zhongjun Ma, Guilin University of Electronic Technology, China; Yu Sun, Jiangsu University, China

This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this paper, we study network coherence characterizing the consensus behaviors with additive noise in a family of book graphs. It is shown that the network coherence is determined by the eigenvalues of the Laplacian matrix. Using the topological structures of book graphs, we obtain recursive relationships for the Laplacian matrix and Laplacian eigenvalues and further derive exact expressions of the network coherence. Finally, we illustrate the robustness of network coherence under the graph parameters and show that the parameters have distinct effects on the coherence.

With the discovery of deterministic small-world [

Calculating the Laplacian spectrum of a network plays an important role in the study of network characteristics. For example, the Kirchhoff index and global mean first-passage time of a network are related to the sum of reciprocals of non-zero eigenvalues [_{2} norm. This concept of the network coherence helps to study the relationship between the Laplacian eigenvalues and network consistency. Great progress has been made for some special networks such as Vicsek fractals [

It is known that the topology of a graph dominates the Laplacian eigenvalues [

The rest of this paper is organized as follows. Book graphs and network coherence are presented in section 2. Section 3 gives detailed calculations of network coherence. Conclusions are given in section 4.

Book graphs _{m} are defined as the graph Cartesian product [_{m} = _{m+1}□_{2}, where _{m}(_{2} is the path graph on two nodes, see _{m,n} of order (_{m,n} = _{m+1}□_{n}, where _{n}(

Book graphs _{m}.

Stacked book graphs _{m,n} with

The network coherence was introduced to characterize the steady-state variance of the deviation from consensus. The relationship [

where _{i}(_{i}(_{i} is the neighboring node set of node _{i}(

Then, the first-order network coherence is defined as the mean, steady-state variance of the deviation from the average of all node values, i.e.,

where

Let 0 = λ_{1} < λ_{2} ≤ … ≤ λ_{N} be the Laplacian eigenvalues. The network coherence is given by

When the network has a smaller variance, it has a higher network coherence, meaning that it is more robust to the noise.

In this section, we present the detailed calculations of the sum of reciprocals of the Laplacian eigenvalues and obtain exact expressions of network coherence. According to the structure of _{m,n}, its Laplacian matrix reads as

where _{m} is the Laplacian matrix of a star graph _{m}, that is,

Then, we need to solve the characteristic equation _{m,n}

where _{i}(1 ≤

Suppose _{m}_{i} = λ_{j}_{i}, _{j}(_{m}. Then, Equation (2) becomes

We then rewrite Equation (3) as

where

Further, we have

We rewrite

From Equation (4), each eigenvalue λ_{j} produces to _{m,n} has _{j} into two cases: λ_{j} ≠ 0 and λ_{j} = 0 to obtain the network coherence.

Let

where the initial conditions are

From Equation (5), we have

where

It follows from Equation (6) that

Solving Equation (7) with initial conditions of

where

Substituting Equation (8) into Equation (6) yields

Next, we need to calculate the first-order terms

where the initial values are

where

We introduce a new polynomial as

Using the Vieta's formula [

When λ_{j} = 0,

Further,

where the initial conditions are

It follows from Equation (9) that

We introduce a polynomial _{n}(λ) to obtain the exact solution of the network coherence, i.e.,

According to Equations (10) and (11), the constant and first-order terms of _{n}(λ) are

Based on the Vieta's theorem [

When _{m} has four eigenvalues, that is, λ_{1} = 0, λ_{2} = λ_{3} = 1, λ_{4} = 4. Using the above-mentioned calculations, we obtain the analytical expression of network coherence, i.e.,

where

To investigate the effect of the parameters

Then, the characteristic polynomial _{m, 2} is

The roots of this polynomial

By the definition (1), we finally obtain the network coherence with regard to the parameters

From the expressions (12) and (13), we plot the relationships between network coherence and the parameters

Network coherence regarding the parameters

In this paper, we have studied the consensus problems in noisy book graphs. Using the graph's constructions, we have obtained the recursive relationships for the Laplacian matrix and Laplacian eigenvalues and proposed a method to derive exact expressions of the sum of reciprocals of these eigenvalues. We then have presented exact solutions of network coherence with regard to graph parameters and investigated their effects on the coherence. It is shown that the larger size of star graphs results in better consensus, while the larger size of path graphs leads to worse consensus. The obtained results showed that the structure difference produces distinct performance on the coherence. Our method for the book graphs could be applied to study their random walks and Kirchhoff index.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

JC, YL, and WS contributed to the conception and design of the study. JC and YL performed the analytical and numerical results. JC and WS wrote the manuscript. All authors contributed to the manuscript revision, read, and approved the submitted version. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.