Network Coherence in a Family of Book Graphs

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.


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 [6][7][8][9], 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 [13][14][15]. 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 H 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 [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, geometricarithmetic 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.

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

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 bẏ where x i (t) is the state of node i and subject to the stochastic noise η i (t). L is the Laplacian matrix. i is the neighboring node set of node i, and η i (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 = 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.

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 B m,n , its Laplacian matrix reads as where L m is the Laplacian matrix of a star graph S m , that is, Then, we need to solve the characteristic equation L m,n x = λx, which is given by . . , n, where λ j (j = 1, 2, . . . , m + 1) are the eigenvalues of L m . Then, Equation (2) becomes We then rewrite Equation (3) as We rewrite R j n (λ) in a recursive form as From Equation (4), each eigenvalue λ j produces to n eigenvalues and B m,n has n(m +1) eigenvalues, denoted by n = {λ n i |1 ≤ i ≤ n(m + 1)} = 1 n ∪ 2 n . . . ∪ m+1 n . For convenient calculations, we denote the smallest eigenvalues λ n 1 = 0. In the following subsections, we divide λ j into two cases: λ j = 0 and λ j = 0 to obtain the network coherence.

Exact Solution of Network Coherence for B m
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 Frontiers in Physics | www.frontiersin.org Then, the characteristic polynomial P(λ) of L m,2 is 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 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) → 1 3 , 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.

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.