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Front. Phys., 16 October 2020 |

Network Coherence in a Family of Book Graphs

Jing Chen1, Yifan Li2 and Weigang Sun2*
  • 1School of Information Technology, Zhejiang Yuying College of Vocational Technology, Hangzhou, China
  • 2School of Sciences, Hangzhou Dianzi University, Hangzhou, China

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.

1. Introduction

With the discovery of deterministic small-world [1] and scale-free [2] networks, deterministically growing network models have gained increasing attention because they can provide exact results for topology and dynamics. As a special type of deterministic networks, fractal networks constructed by fractal structures, such as Koch fractals [3], Sierpinski fractals [4], and Vicsek fractals [5], have been widely studied. Presently the main issues that require consideration in fractal networks include random walks [69], consensus dynamics [10, 11] and percolation [12]. It is proved that fractal networks are good candidate network models for verifying the results of random graphs.

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 [1315]. The synchronizability [16] of a network refers to the ratio of the second smallest eigenvalue to the largest eigenvalue of the Laplacian matrix. In addition, the effective graph resistance is connected with the Laplacian spectrum [17]. Recently, network coherence [10] was introduced to characterize the extent of consensus of coupled agents under the noisy circumstance and was determined by the Laplacian spectrum in an H2 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 [10], tree-like networks [11], Sierpiński graphs [18] and weighted networks [19]. Many works have been devoted to studying the network coherence. Hong et al. studied the role of Laplacian energy on the coherence in a family of tree-like networks with controlled initial states [20]. Patterson and Bamieh investigated the leader-follower coherence and proposed optimal algorithms to select the leaders [21]. Later, Sun et al. proposed a leader centrality to identify more influential spreaders using the optimal coherence [22].

It is known that the topology of a graph dominates the Laplacian eigenvalues [23]. Thus, calculating the Laplacian eigenvalues is a technical challenge and it is theoretical and practical interest to find new ways to calculate them. In this paper, a family of book graphs is chosen as our network models. The topological indices, e.g., randic index, sum connectivity index, geometric-arithmetic index, fourth atom-bond connectivity index, and edge labeling, have been analytically obtained [24, 25]. However, the dynamics of the book graphs remains less understood, in spite of the facts that studying the dynamical processes leads to a better understanding of how the underlying systems work.

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.

2. Model Presentation and Network Coherence

2.1. Book Graphs

Book graphs Bm are defined as the graph Cartesian product [26], i.e., Bm = Sm+1P2, where Sm(m ≥ 1) is a star graph and P2 is the path graph on two nodes, see Figure 1. The stacked book graphs Bm,n of order (m, n) are Bm,n = Sm+1Pn, where Pn(n ≥ 2) is the path graph on n nodes, see Figure 2.


Figure 1. Book graphs Bm.


Figure 2. Stacked book graphs Bm,n with m = 3, n = 6.

2.2. Network Coherence

The network coherence was introduced to characterize the steady-state variance of the deviation from consensus. The relationship [10] between network coherence and Laplacian eigenvalues was established. The consensus dynamics with the additive noise are given by


where xi(t) is the state of node i and subject to the stochastic noise ηi(t). L is the Laplacian matrix. Ωi is the neighboring node set of node i, and ηi(t) is a delta-correlated Gaussian noise.

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 var is the expectation of the squared deviation of a random variable from its mean.

Let 0 = λ1 < λ2 ≤ … ≤ λN be the Laplacian eigenvalues. The network coherence is given by

H=12Ni=2N1λi.    (1)

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

3. Calculations of Network Coherence

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 Bm,n, its Laplacian matrix reads as


where Lm is the Laplacian matrix of a star graph Sm, that is,


Then, we need to solve the characteristic equation Lm,nx = λx, which is given by

 (Lm+Im+1)x1Im+1x2=λx1,Im+1x1+(Lm+2Im+1)x2Im+1x3=λx2,     Im+1xn1+(Lm+Im+1)xn=λxn,    (2)

where x=(x1T,x2T,,xnT)T and the dimension of xi(1 ≤ in) is m + 1.

Suppose Lmxi = λjxi, i = 1, 2, …, n, where λj(j = 1, 2, …, m + 1) are the eigenvalues of Lm. Then, Equation (2) becomes

(λj+1)x1x2=λx1,x1+(λj+2)x2x3=λx2,      xn1+(λj+1)xn=λxn.    (3)

We then rewrite Equation (3) as




Further, we have

Rnj(λ)=λj,j=1,2,,m+1.    (4)

We rewrite Rnj(λ) in a recursive form as


From Equation (4), each eigenvalue λj produces to n eigenvalues and Bm,n has n(m + 1) eigenvalues, denoted by Λn={λin|1in(m+1)}=Λn1Λn2Λnm+1. For convenient calculations, we denote the smallest eigenvalues λ1n=0. In the following subsections, we divide λj into two cases: λj ≠ 0 and λj = 0 to obtain the network coherence.

3.1. When λj ≠ 0, j = 2, …, m + 1

Let Rnj(λ)=Tnj(λ)/Pnj(λ), where Tnj(λ) and Pnj(λ) are two polynomials satisfying gcd[Tnj(λ),Pnj(λ)]=1, the term gcd is the greatest common divisor. Then, we obtain the following recursive relationships as

Tnj(λ)=[Tn-1j(λ)-(λj+1)Pn-1j(λ)]λ-Tn-1j(λ)+λjPn-1j(λ),Pnj(λ)=Tn-1j(λ)-(λj+1)Pn-1j(λ),    (5)

where the initial conditions are


From Equation (5), we have

{tnj(0)=-tn-1j(0)+λjpn-1j(0),pnj(0)=tn-1j(0)-(λj+1)pn-1j(0).    (6)

where tnj(0) and pnj(0) are the constant terms of Tnj(λ) and Pnj(λ).

It follows from Equation (6) that

pnj(0)+(λj+2)pn-1j(0)+pn-2j(0)=0.    (7)

Solving Equation (7) with initial conditions of p2j(0)=-(λj+1) and p3j(0)=λj2+3λj+1 yields

pnj(0)=c1j(r1j)g+c2j(r2j)g,    (8)

where r1j and r2j are the roots of the characteristic equation λ2+(λj+2)λ+1=0. The constants r1j,r2j,c1j and c2j are


Substituting Equation (8) into Equation (6) yields


Next, we need to calculate the first-order terms tnj(1),pnj(1) of Tnj(λ) and Pnj(λ). Using the relationship between Tnj(λ) and Pnj(λ) of Equation (5) gives


where the initial values are t2j(1)=-(λj+2),p2j(1)=1,p3j(1)=-(2λj+3). Then, we obtain

tnj(1)={ej(r1j)2+[ngj+(λj+1)ej]r1j            +(n1)(λj+1)gj}(r1j)n2            +{fj(r2j)2+[nhj+(λj+1)fj]r2j            +(n1)(λj+1)hj}(r2j)n2,pnj(1)=ej(r1j)n1+fj(r2j)n1+(n1)[gj(r1j)n2+hj(r2j)n2],



We introduce a new polynomial as

Dnj(λ)=Tnj(λ)-λjPnj(λ),               =(λ-λ(j-1)n+1n)(λ-λ(j-1)n+2n)(λ-λjnn).    (9)

Using the Vieta's formula [26, 27] for Dnj(λ)=0, we obtain its constant and first-order terms, denoted by dnj(0),dnj(1), that is,

{dnj(0)=tnj(0)-λjpnj(0)            =-c1j(r1j)n-2[1+(1+λj)r1j]            -c2j(r2j)n-2[1+(1+λj)r2j],dnj(1)=tnj(1)-λjpgj(1)            =(r1j)n-2[ej(r1j)2+(ngj+ej)r1j+(n-1)gj]               +(r2j)n-2[fj(r2j)2+(nhj+fj)r2j+(n-1)hj].    (10)

3.2. When λj = 0

When λj = 0, Rn1(λ)=0 has only one root λ1n=0. To obtain all the non-zero roots of Rnj(λ)=0, we introduce a new polynomial, i.e.,




where the initial conditions are T21(λ)=λ-2,P21(λ)=λ-1. In the same way, we obtain the following coefficients, which are given by


It follows from Equation (9) that

{dn1(0)=(1)n1λ2nλ3nλnn, =(1)n1n,dn1(1)=(1)n2[λ3nλ4nλnn+λ2nλ4nλnn+ +λ2nλ3nλn1n], =(1)n2·n(n21)6.    (11)

3.3. Exact Solution of Network Coherence for Bm,n

We introduce a polynomial Dn(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 Dn(λ) are

dn(0)=j=1m+1dnj(0),dn(1)=dn1(1)dn2(0)dnm+1(0)m+1                +dn1(0)dn2(1)dnm+1(0)m+1++dn1(0)dn2(0)dnm+1(1)m+1.

Based on the Vieta's theorem [26, 27], the network coherence reads as


When m = 3, the Laplacian matrix Lm has four eigenvalues, that is, λ1 = 0, λ2 = λ3 = 1, λ4 = 4. Using the above-mentioned calculations, we obtain the analytical expression of network coherence, i.e.,

H(n)=18n{n216               20[α1+g2(nr12+n1)](r12)n2+[α2+h2(nr22+n1)](r22)n2β1(r12)n2+β2(r22)n2              2[θ1+g4(nr14+n1)](r14)n2+[θ2+h4(nr24+n1)](r24)n2η1(r14)n2+η2(r24)n2},    (12)

where α1=-5-2525, α2=-5+2525, β1=15-75,β2=15+75, θ1=-10-7232, θ2=-10+7232, η1=24-172, η2=24+172, g2=(55)(3r12+1))10r12(2r12+3), h2=(5+5)(3r22+1))10r22(2r22+3), g4=(22)(6r14+1))8r14(r14+3), h4=(2+2)(6r24+1))8r24(r24+3), r12=-3+52, r22=-3-52, r14=-3+22, r24=-3-22.

3.4. Exact Solution of Network Coherence for Bm

To investigate the effect of the parameters m on the network coherence, we propose another method to obtain the solution regarding the parameters m. When n = 2, the Laplacian matrix is


Then, the characteristic polynomial P(λ) of Lm, 2 is

P(λ)=|Lm-λIm+1-Im+1-Im+1Lm-λIm+1|           =|Lm-(λ+1)I|·|Lm-(λ-1)I|           =λ(λ-2)(λ-m-1)(λ-m-3)(λ-1)m-1(λ-3)m-1.

The roots of this polynomial P(λ) are as follows,


By the definition (1), we finally obtain the network coherence with regard to the parameters m, which is given by

H(m)=14m+4[12+1m+1+1m+3+4(m-1)3].    (13)

From the expressions (12) and (13), we plot the relationships between network coherence and the parameters m and n, see Figure 3. It shows that the values of network coherence linearly increase with n, while the network coherence will achieve a steady constant state for a large m, i.e., H(m)13, meaning that the consensus displays worse with increasing values of n. In a word, the number of nodes n in the path graph has more influence than the number of nodes m in the star graph.


Figure 3. Network coherence regarding the parameters n and m.

4. Conclusions

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.

Data Availability Statement

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

Author Contributions

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.


This work was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY20F030007).

Conflict of Interest

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.


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Keywords: consensus, coherence, book graph, Laplacian spectra, recursive

Citation: Chen J, Li Y and Sun W (2020) Network Coherence in a Family of Book Graphs. Front. Phys. 8:583603. doi: 10.3389/fphy.2020.583603

Received: 15 July 2020; Accepted: 08 September 2020;
Published: 16 October 2020.

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

Copyright © 2020 Chen, Li and Sun. 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.

*Correspondence: Weigang Sun,