First- and second-order network coherence in -duplication weighted corona networks

This paper studies first- and second-order coherence problems for $N$-duplication weighted corona networks subject to stochastic disturbances. Explicit expressions of the coherence for first-order (and second-order) dynamics, which are determined by the sum of the reciprocal (and square of reciprocal) of each nonzero eigenvalue of the Laplacian matrix, are derived. In particular, for both first- and second-order systems, the analytical formulas of the network coherence are presented from two different perspectives. Based on these formulas, the influence of the duplication $N$, the weight $\omega$, and the factor networks $G_1$ and $G_2$ on the network coherence of the corona network $G_1◦G_2$ is investigated. Some noteworthy topological properties of the $N$-duplication weighted corona network are also revealed.

1 Introduction

Over the past few years, technological development in communication networks has greatly increased interest in distributed coordination for networks of dynamical agents. As one of the fundamental problems in cooperative control, the consensus problem for multi-agent systems has been investigated from various perspectives [1–5]. In the context of networks (graphs) of agents, consensus means that agents represented by nodes (vertices) reach an agreement on a certain issue, such as pace, load, or direction and velocity.

In realistic applications, communication between agents is often degraded due to environmental uncertainty or communication uncertainty, for example, thermal, fading channel, and quantization noises during encoding and decoding. Without uncertainties, it is well known that when the graph is connected, the states of autonomous systems converge exponentially to the average of the initial state values. In the presence of stochastic disturbances, however, the state evolution becomes a stochastic process and fluctuates around the average of the current node states. Thus, it is of great interest to consider how robust distributed consensus algorithms are to external disturbances [6–16]. Network coherence [8–16] quantifies the steady-state variance of these fluctuations and can be considered a measure of the robustness of the consensus process to the additive noise. Networks with small steady-state variance have high network coherence and can be considered to be more robust to noise than networks with low coherence [13–16]. For both first- and second-order systems, network coherence can be measured by the $H_2$-norm of the consensus errors, which can be characterized by the spectrum of the Laplacian matrix of the underlying communication graph [15, 16].

Because massive networks often consist of small pieces, for example, communities [17] and motifs [18], graph products, by which one can build a large network out of two or more smaller ones, are widely used as an effective method for generating large-scale networks. Analysis of product networks offers critical insights into understanding the dynamics of real-world large-scale networks. Specifically, graph products have been explored to construct and reveal structural and functional relationships between factor systems and the associated composite system [19–22].

Common graph products include direct products and strong products [23], Cartesian products [24], Kronecker products [25], and corona products [26]. Among them, corona product graphs have attracted a great deal of attention due to their complex but unique structures and wide range of applications in coding theory, DNA sampling, UAV formation, and some special chemical and biological structures or communities [27–34]. The concept of the corona product of graphs was first introduced by Frucht and Harary [27]. In [28], the authors introduced the edge corona of graphs and calculated the corresponding spectrum. The properties of spectra and Laplacian spectra of corona products have been extensively studied in spectral graph theory [35–38]. Some recently widely concerned indices, such as the Sombor index and the Kirchhoff index, have been derived from corona product graphs [39, 40]. Notably, some related advancements in graph theory have been reported in [41–45]. In the literature, based on the spectral analysis of the Laplacian and normalized Laplacian matrices, Kemeny’s constant, global mean-first-passage time of random walks, and the number of spanning trees were studied in various network structures. Specifically, Kemeny’s constant represents the cumulative sum of relaxation time scales and has specific applications in computing a graph’s Kirchhoff index. These research achievements and methods are enlightening for further studies on corona networks.

The first-order coherence of weighted corona networks was examined from the weighted Laplacian spectra perspective in [46]. It is noteworthy that, in addition to the basic corona product investigated in [46], multiple variants of corona operation have been introduced and studied, including edge corona [35], neighborhood corona [36], subdivision double corona, Q-graph double corona, R-graph double corona [37], and iterative corona [38], etc. Therefore, further research on consensus algorithms of various kinds of corona networks is necessary. Moreover, little research has clearly addressed the relationship between the coherence of the corona network and that of its factor networks, and research results on the second-order or higher-order coherence of corona networks are still rare.

The multilayer network is a frontier research branch of network science. The multilayered structure has many examples in reality, for instance, the interactions between the power grid and the Internet, friendship and family relations, or transportation and aviation networks [47, 48]. Lately, a multilayered graph based on the duplication of corona products was introduced in [49, 50]. Varghese and Susha [49] determined the Laplacian spectrum and discussed the number of spanning trees, the Kirchhoff index, and the incidence energy of the graph. The controllability of the $N$-duplication corona product network $G_1◦G_2$ was investigated in [50]. An example of this $N$-duplication corona network is shown in Figure 1, where $G_1$ and $G_2$ are complete graphs of order 5 and 2, respectively.

Figure 1

Figure 1. Illustration of $N$-duplication corona graphs $G_1◦G_2$ composed of $G_1$ and $G_2$, where (a) and (b) indicate 1-duplication and 2-duplication corona graphs, respectively.

With the introduction of duplication, the classic corona graph is generalized from a single-layer structure to a multilayered structure. It is necessary and significant to extend the consensus theory to the $N$-duplication corona product network, which includes the basic corona network as a special case with the duplication $N=1$. The intricate topological configurations of the $N$-duplication corona product network not only compounds analytical challenges in coherence studies but also raises new research questions. What influence will the duplication $N$ or the weight factor $\omega$ have on the network coherence? What is the relationship between the coherence of the corona network $G_1◦G_2$ and that of its factor networks $G_1$ and $G_2$? In comparison, which factor network plays a more important role in the determination of the coherence of the composite corona network? These natural and interesting questions deserve to be considered.

Inspired by these questions, this study explores the robustness of consensus algorithms for $N$-duplication weighted corona networks when the nodes are subject to external perturbations. The coherence in corona product networks composed of first- or second-order dynamic agents is studied, aiming to obtain exact solutions of network coherence and unveil the relationship between the network topology and network coherence. The main contributions of this work are three-fold. First, the explicit expressions for the coherence of the first- and second-order noisy consensus algorithms in $N$-duplication weighted corona networks are obtained. The results of [46] are now a special case of this study, and more detailed and noteworthy analysis is presented in this work. Second, for both first- and second-order consensus algorithms, the impacts of the duplication $N$ and the weight $\omega$ on the network coherence are explored. It is found that corona networks with larger duplication $N$ or higher weight $\omega$ usually have higher network coherence and can be considered to be more robust to noise. Note that the property of high coherence of the network with large duplication $N$ can be regarded as a special topological characteristic of the $N$-duplication weighted corona network. Finally, based on the obtained formulas, the relationship between the coherence of the composite corona network $G_1◦G_2$ and that of the factor networks $G_1$ and $G_2$ is investigated. Little work has been done from this perspective on the study of coherence problems. It is revealed that, for both first- and second-order consensus algorithms, higher coherence of $G_1$ or $G_2$ usually also leads to higher coherence of the corona network $G_1◦G_2$. Especially in the situation of large duplication $N$ or high weight $\omega$, the network coherence of the corona network $G_1◦G_2$ is mainly determined by the factor network $G_1$. The results presented in this study not only contribute to the theoretical understanding of network coherence but also provide practical insights into how different parameters and network structures can be optimized for better coherence, which is crucial for the design and analysis of complex networks in various applications such as sensor networks, social networks, and biological networks.

The rest of this paper is organized as follows. Section 2 reviews the definition of $N$-duplication weighted corona network, the first- and second-order network coherence, and the relationship between the network coherence and the Laplacian spectrum of the underlying graph. Section 3 shows the explicit analytical results of first- and second-order network coherence in $N$-duplication weighted corona networks. The influence of the duplication $N$, the weight $\omega$, and the factor networks $G_1$ and $G_2$ on the network coherence of the corona network $G_1◦G_2$ is investigated. Several simulation examples are presented in Section 4. Finally, Section 5 draws the conclusion.

2 Notations and preliminaries

This section briefly reviews the definition of the first- and second-order network coherence and the formation of $N$-duplication weighted corona networks and introduces some lemmas that will be used in the sequel.

2.1 Notations

Some available notations used in this study are given in the abbreviations.

2.2 Coherence in networks with first-order dynamics

Consider a network with underlying undirected graph $G=(V,E,A)$, where $V=\{v_1,\dots ,v_n\}$ is the set of nodes, $E\subseteq V\times V$ is the set of edges, and $A=[a_{ij}]\in R^{n\times n}$ is the weighted adjacency matrix. The Laplacian matrix $L$ is defined as $L=D-A$, where $D$ is the degree matrix. The network has consensus dynamics modeled by the stochastic differential equation

$\dot{x}\left(t\right)=-Lx\left(t\right)+\chi \left(t\right),(1)$

where $x(t)\in R^n$ is the state of the system, $L$ is the Laplacian matrix, and $\chi (t)$ is an $n$-vector of zero-mean and unit variance white noise. The network coherence of the first-order system in Equation 1 is denoted by the mean steady-state variance of the deviation from the average of all node values [7–9], that is,

$H_1\left(G\right)≔\underset{t\to \infty }{\lim }\frac{1}{n}{\displaystyle \sum _{i=1}^n}var\left\{x_i\left(t\right)-\frac{1}{n}{\displaystyle \sum _{j=1}^n}{x}_j\left(t\right)\right\}.$

It has been shown [6–9] that $H_1(G)$ can be completely determined by the eigenvalues of the Laplacian matrix $L$,

$H_1\left(G\right)=\frac{1}{2n}{\displaystyle \sum _{i=2}^n}\frac{1}{{\lambda }_i},(2)$

where $0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_n$ are the Laplacian eigenvalues.

2.3 Coherence in networks with second-order dynamics

In the second-order system, such as the vehicle formation problem, the node states consist of a position vector $x(t)\in R^n$ and a velocity vector $v(t)\in R^n$. The second-order consensus dynamics subject to noise are described by

$\left[\begin{array}{c}\hfill \dot{x}\left(t\right)\hfill \\ \hfill \dot{v}\left(t\right)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 0\hfill & \hfill I\hfill \\ \hfill -L\hfill & \hfill -L\hfill \end{array}\right]\left[\begin{array}{c}\hfill x\left(t\right)\hfill \\ \hfill v\left(t\right)\hfill \end{array}\right]+\left[\begin{array}{c}\hfill 0\hfill \\ \hfill I\hfill \end{array}\right]\omega \left(t\right),(3)$

where $\omega (t)$ is a disturbance vector with zero-mean and unit variance.

The network coherence of the second-order system in Equation 3 is defined in terms of $x(t)$ only and denoted by the mean (over all nodes) and the steady-state variance of the deviation from the average of $x(t)$,

$H_2\left(G\right)≔\underset{t\to \infty }{\lim }\frac{1}{n}{\displaystyle \sum _{i=1}^n}var\left\{x_i\left(t\right)-\frac{1}{n}{\displaystyle \sum _{j=1}^n}{x}_j\left(t\right)\right\}.$

This value is also completely determined by the eigenvalues of the Laplacian matrix [8, 9, 15],

$H_2\left(G\right)=\frac{1}{2n}{\displaystyle \sum _{i=2}^n}\frac{1}{{\left({\lambda }_i\right)}^2},(4)$

where $0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_n$ are the Laplacian eigenvalues.

Note that networks with smaller steady-state variance $H_1(G)$ or $H_2(G)$ have higher network coherence and can be considered to be more robust to noise than networks with lower coherence [11–14].

The following lemma gives the well-known Vieta’s formulas, which will be leveraged in the following coherence analysis.

Lemma 1. [52] Let $p(x)=a_n{x}^n+\cdots +a_{1}x+a_0$ be a real polynomial of degree $n\ge 2$ with $a_0\ne 0$. It shows

${\displaystyle \sum _{k=1}^n}\frac{1}{{\rho }_k}=-\frac{a_1}{a_0},$

and

${\displaystyle \sum _{k=1}^n}\frac{1}{{\rho }_k^2}={\left(\frac{a_1}{a_0}\right)}^2-2\frac{a_2}{a_0},$

where ${\rho }_k(1\le k\le n)$ are the roots of $p(x)$.

2.4 $N$-duplication weighted corona networks

As an extension of the classic corona network, an $N$-duplication weighted corona network was recently introduced in the literature [49, 50].

Definition 1. [49, 50] Let $G_1$ and $G_2$ be two finite, simple, nonempty, and vertex-disjoint graphs with $n_1$ and $n_2$ vertices, respectively. The $N$-duplication weighted corona product $G_1◦G_2$ is generated by taking $N$ copies of $G_1$ and $n_1$ copies of $G_2$ and then joining the $i$th vertex of each $G_1$ to every vertex in the $i$th copy of $G_2$ $(i=\mathrm{1,2,3},\dots ,n_1)$. All the weights of the newly added edges between $G_1$ and $G_2$ are the same and positive, denoted as $\omega$.

Let $L_1$ and $L_2$ be the Laplacian matrices of $G_1$ and $G_2$ respectively, then the Laplacian matrix of the $N$-duplication weighted corona product $G_1◦G_2$ is

$L=\left[\begin{array}{cc}\hfill I_N\otimes \left(L_1+n_2\omega I_{n_1}\right)\hfill & \hfill -1_N\otimes \left(\omega I_{n_1}\otimes 1_{n_2}^T\right)\hfill \\ \hfill -1_N^T\otimes \left(\omega I_{n_1}\otimes 1_{n_2}\right)\hfill & \hfill I_{n_1}\otimes \left(L_2+N\omega I_{n_2}\right)\hfill \end{array}\right].(5)$

The join operation of two disjoint graphs is also an effective method for generating large-scale networks [51]. As an extension, [46] presented the weighted join operation as below.

Definition 2. [46, 51] The join of two disjoint graphs $G_1$ and $G_2$, denoted by $G_1\vee G_2$, is the graph with vertex set $V(G_1)\bigcup V(G_2)$ and the edge set $E(G_1)\bigcup E(G_2)\bigcup \{(u,v),foreachu\in V(G_1)andv\in V(G_2)\}$. $(u,v)$ represents the added edge joining $u$ and $v$. Each link $(u,v)$ has the same and positive weight, called the join-weight of $G_1\vee G_2$.

In [49, 50], the characteristic polynomial of the $N$-duplication weighted corona graph $G_1◦G_2$ was presented based on the eigenvalues of the factor graphs $G_1$ and $G_2$.

Lemma 2. [49, 50] Let $\sigma (G_1)=\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{n_1}|0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_{n_1}\}$ and $\sigma (G_2)=\{{\mu }_1,{\mu }_2,\dots ,{\mu }_{n_2}|0={\mu }_1<{\mu }_2\le \cdots \le {\mu }_{n_2}\}$ be the Laplacian spectrum sets of $G_1$ and $G_2$, respectively. Then, the Laplacian characteristic polynomial of the $N$-duplication weighted corona product $G_1◦G_2$ with the Laplacian matrix Equation 5 is

$\begin{array}{cc}\mathrm{\Phi }\left(L:\gamma \right)=\hfill & \gamma \left(\gamma -\left(N\omega +n_2\omega \right)\right)\prod _{j=2}^{n_2}{\left(\gamma -N\omega -{\mu }_j\right)}^{n_1}\prod _{i=1}^{n_1}{\left(\gamma -n_2\omega -{\lambda }_i\right)}^{N-1}\hfill \\ \hfill & \prod _{i=2}^{n_1}\left({\gamma }^2-\left(N\omega +n_2\omega +{\lambda }_i\right)\gamma +N\omega {\lambda }_i\right).\hfill \end{array}(6)$

3 Main results

This section studies the first- and second-order coherence problems for $N$-duplication weighted corona networks, where vertices are subject to white noise. Note that the graphs $G_1$, $G_2$ and the $N$-duplication weighted corona network $G_1◦G_2$ used in the sequel are all as defined in Def. 1.

3.1 First-order coherence of the $N$-duplication weighted corona network

Theorem 1. Let $\sigma (G_1)=\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{n_1}|0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_{n_1}\}$ and $\sigma (G_2)=\{{\mu }_1,{\mu }_2,\dots ,{\mu }_{n_2}|0={\mu }_1<{\mu }_2\le \cdots \le {\mu }_{n_2}\}$ be the Laplacian spectrum sets of $G_1$ and $G_2$, respectively. Then the first-order coherence of the $N$-duplication weighted corona network $G_1◦G_2$ can be described as follows:

$\begin{array}{cc}{H}_1\left(G_1◦G_2\right)=\hfill & \frac{1}{2n_{1}N}{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i}+\frac{N-1}{2n_1\left(n_2+N\right)}{\displaystyle \sum _{i=1}^{n_1}}\frac{1}{n_2\omega +{\lambda }_i}\hfill \\ \hfill & +\frac{1}{2\left(n_2+N\right)}{\displaystyle \sum _{j=2}^{n_2}}\frac{1}{N\omega +{\mu }_j}+\frac{1}{2n_1{\left(n_2+N\right)}^2\omega }+\frac{n_1-1}{2n_1\left(n_2+N\right)N\omega }.\hfill \end{array}(7)$

Proof. By Def. 1, the $N$-duplication weighted corona network $G_1◦G_2$ contains $n_1(n_2+N)$ vertices. From Lem. 2, the Laplacian eigenvalues of $G_1◦G_2$ are the roots of $\mathrm{\Phi }(L:\gamma )=0$. Then, the Laplacian spectrum of $G_1◦G_2$ consists of

(i) 0;

(ii) $N\omega +n_2\omega$;

(iii) $N\omega +{\mu }_j$, repeated $n_1$ times for $j=\mathrm{2,3},\dots ,n_2$;

(iv) $n_2\omega +{\lambda }_i$, repeated $N-1$ times for $i=\mathrm{1,2},\dots ,n_1$;

(v) Two roots of the equation

${\gamma }^2-\left(N\omega +n_2\omega +{\lambda }_i\right)\gamma +N\omega {\lambda }_i=0,i=\mathrm{2,3},\dots ,n_1.$

From Equation 2, the first-order network coherence of $G_1◦G_2$ is determined by the sum of the inverses of nonzero Laplacian eigenvalues. Consider the last factor of the Laplacian characteristic polynomial defined in Equation 6:

$\prod _{i=2}^{n_1}\left({\gamma }^2-\left(N\omega +n_2\omega +{\lambda }_i\right)\gamma +N\omega {\lambda }_i\right).(8)$

Let $a_0$ and $a_1$ denote the constant term and the coefficient of the linear term of Equation 8, respectively. Then,

$\begin{array}{c}{a}_0=\prod _{i=2}^{n_1}\left(N\omega {\lambda }_i\right),\hfill \\ a_1=-\prod _{i=2}^{n_1}\left(N\omega {\lambda }_i\right){\displaystyle \sum _{i=2}^{n_1}}\left[\left(N\omega +n_2\omega +{\lambda }_i\right)/\left(N\omega {\lambda }_i\right)\right],\hfill \end{array}(9)$

and

$-\frac{a_1}{a_0}={\displaystyle \sum _{i=2}^{n_1}}\frac{N\omega +n_2\omega +{\lambda }_i}{N\omega {\lambda }_i}=\frac{n_1-1}{N\omega }+\frac{N+n_2}{N}{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i}.(10)$

Combining the Laplacian spectrum of $G_1◦G_2$ as shown in (i)-(v), Lemma 1, and Equations 2, 10, one obtains

$\begin{array}{cc}{H}_1\left(G_1◦G_2\right)\hfill & =\frac{1}{2n_1\left(n_2+N\right)}\left(\frac{1}{\left(n_2+N\right)\omega }+\frac{n_1-1}{N\omega }+\frac{N+n_2}{N}{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i}+{\displaystyle \sum _{i=1}^{n_1}}\frac{N-1}{n_2\omega +{\lambda }_i}+{\displaystyle \sum _{j=2}^{n_2}}\frac{n_1}{N\omega +{\mu }_j}\right)\hfill \\ \hfill & =\frac{1}{2n_{1}N}{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i}+\frac{N-1}{2n_1\left(n_2+N\right)}{\displaystyle \sum _{i=1}^{n_1}}\frac{1}{n_2\omega +{\lambda }_i}+\frac{1}{2\left(n_2+N\right)}{\displaystyle \sum _{j=2}^{n_2}}\frac{1}{N\omega +{\mu }_j}\hfill \\ \hfill & +\frac{1}{2n_1{\left(n_2+N\right)}^2\omega }+\frac{n_1-1}{2n_1\left(n_2+N\right)N\omega }.\hfill \end{array}$

The proof is completed.

Remark 1. From Equation 7, Theorem 1 implies that $H_1(G_1◦G_2)$ decreases as the duplication $N$ or the weight factor $\omega$ increases. Accordingly, corona networks $G_1◦G_2$ with larger duplication $N$ or weight factor $\omega$ have higher first-order network coherence and can be considered to be more robust to noise than networks with smaller duplication or weight factors.

It is worth noting that the phenomenon of high coherence of the corona network $G_1◦G_2$ with a large duplication $N$ is interesting because it differs from the results reported in prior literature, such as [14, 15]. [14] considered the first-order network coherence in a kind of 5-rose graph and found that 5-rose networks with small network sizes have high network coherence. In [15], the authors investigated the coherence problem of the Koch network and revealed that enhancing the iteration or the network size of the Koch network will reduce the network coherence. Thus, high coherence or strong robustness of the corona network $G_1◦G_2$ with a large duplication $N$ (thereby large network size) can be regarded as a distinctive topological characteristic that may lead to significant application value. For example, in [8], the vehicular formation control problem was studied based on the analysis of performance measures in large-scale networks. It is found that the network coherence, which varies with network size and dimension, plays an important role in the performance limitation of the vehicle formation. From this point of view, the $N$-duplication weighted corona network $G_1◦G_2$ can be considered a graph with good robustness to external disturbances, which provides new insights into its practical applications.

In Equation 7, $H_1(G_1◦G_2)$ is characterized by the Laplacian eigenvalues of the associated matrices $G_1$ and $G_2$, the weight $\omega$, and the duplication $N$. To further explore the relationship between the first-order coherence of the composite network $G_1◦G_2$ and that of the factor networks $G_1$ and $G_2$, we derive another analytical formula for $H_1(G_1◦G_2)$.

Lemma 3. Let $G$ be a simple graph with $n$ vertices and Laplacian eigenvalues $0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_n$. There is an orthogonal matrix $P={(p_{ij})}_{n\times n}$, such that $P^{T}L(G)P=diag({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$. Moreover, $p_1=\sqrt{n},p_2=\cdots =p_n=0$, where $p_i(i=\mathrm{1,2},\dots ,n)$ is the sum of the $i$th column of the matrix $P$.

Proof. For the Laplacian matrix $L(G)$, it is obvious that ${\xi }_1=\frac{1}{\sqrt{n}}{(\mathrm{1,1},\dots ,1)}^T$ is the unit eigenvector associated with eigenvalue ${\lambda }_1=0$. Let ${\xi }_i(1<i\le n)$ be an eigenvector of $L(G)$ associated with eigenvalue ${\lambda }_i$. Then, ${\xi }_1^T{\xi }_i=0$; that is to say, the sum of all the entries of ${\xi }_i(1<i\le n)$ is 0. The conclusion of the lemma follows from the orthogonal decomposition theorem.

Theorem 2. Let $k_1$ denote the complete graph of order 1 (i.e., the trivial graph), $G_1\vee k_1$ be the join graph of $G_1$ and $k_1$ with the join-wight $n_2\omega$, and $G_2\vee k_1$ be the join graph of $G_2$ and $k_1$ with the join-wight $N\omega$. Then, the first-order coherence of the $N$-duplication weighted corona network $G_1◦G_2$ can be described as follows:

$\begin{array}{cc}{H}_1\left(G_1◦G_2\right)=\hfill & \frac{1}{N}{H}_1\left(G_1\right)+\frac{\left(N-1\right)\left(n_1+1\right)}{n_1\left(n_2+N\right)}{H}_1\left(G_1\vee k_1\right)+\frac{n_2+1}{n_2+N}{H}_1\left(G_2\vee k_1\right)\hfill \\ \hfill & +\frac{1}{2\left(n_2+N\right)\omega }\left[\frac{n_1-1}{N{n}_1}+\frac{1}{n_1\left(n_2+N\right)}+\frac{N-1}{n_2\left(n_1+1\right)}-\frac{1}{N\left(n_2+1\right)}\right].\hfill \end{array}(11)$

Proof. Let $A_1$ and $L_1$ denote the adjacency and Laplacian matrices of $G_1$, respectively. The block form of the adjacency matrix of $G_1\vee k_1$ is

$A\left(G_1\vee k_1\right)=\left[\begin{array}{cc}\hfill A_1\hfill & \hfill n_2\omega 1_{n_1}\hfill \\ \hfill n_2\omega 1_{n_1}^T\hfill & \hfill 0\hfill \end{array}\right],$

where $1_{n_1}$ represents the all-ones column vector of dimension $n_1$. The Laplacian matrix of $G_1\vee k_1$ is

$L\left(G_1\vee k_1\right)=\left[\begin{array}{cc}\hfill L_1+n_2\omega I_{n_1}\hfill & \hfill -n_2\omega 1_{n_1}\hfill \\ \hfill -n_2\omega 1_{n_1}^T\hfill & \hfill n_2\omega \hfill \end{array}\right].$

Suppose that $\sigma (G_1)=\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{n_1}|0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_{n_1}\}$ is the spectrum set of $G_1$. Then, there is an orthogonal matrix $P$ such that $P^T{L}_{1}P=diag({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{n_1})$. In addition,

$\begin{array}{ll}\hfill & \left[\begin{array}{cc}\hfill P^T\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill L_1+n_2\omega I_{n_1}\hfill & \hfill -n_2\omega 1_{n_1}\hfill \\ \hfill -n_2\omega 1_{n_1}^T\hfill & \hfill n_2\omega \hfill \end{array}\right]\left[\begin{array}{cc}\hfill P\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{cc}\hfill P^T\left(L_1+n_2\omega I_{n_1}\right)P\hfill & \hfill -n_2\omega P^T{1}_{n_1}\hfill \\ \hfill -n_2\omega 1_{n_1}^{T}P\hfill & \hfill n_2\omega \hfill \end{array}\right].\hfill \end{array}$

From Lemma 3, the characteristic polynomial of $L(G_1\vee k_1)$ is

$\begin{array}{c}|\lambda I_{n_1+1}-L\left(G_1\vee k_1\right)|=\left|\begin{array}{cc}\hfill P^T\left(\lambda I_{n_1}-L_1-n_2\omega I_{n_1}\right)P\hfill & \hfill n_2\omega P^T{1}_{n_1}\hfill \\ \hfill n_2\omega 1_{n_1}^{T}P\hfill & \hfill \lambda -n_2\omega \hfill \end{array}\right|\hfill \\ =\left|\begin{array}{cccc}\hfill \lambda -n_2\omega \hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill n_2\omega \sqrt{n_1}\hfill \\ \hfill 0\hfill & \hfill \lambda -n_2\omega -{\lambda }_2\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill n_2\omega \sqrt{n_1}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \lambda -n_2\omega \hfill \end{array}\right|.\hfill \end{array}$

Therefore, the Laplacian eigenvalues of $G_1\vee k_1$ with the join-weight $n_2\omega$ are

$0,{\lambda }_i+n_2\omega \left(i=2,\dots ,n_1\right),\left(n_1+1\right)n_2\omega .(12)$

Similarly, the Laplacian eigenvalues of $G_2\vee k_1$ with the join-weight $N\omega$ are

$0,{\mu }_i+N\omega \left(i=2,\dots ,n_2\right),\left(n_2+1\right)N\omega .(13)$

Moreover, we have

$H_1\left(G_1\vee k_1\right)=\frac{1}{2\left(n_1+1\right)}\left({\displaystyle \sum _{i=2}^{n_1}}\frac{1}{n_2\omega +{\lambda }_i}+\frac{1}{\left(n_1+1\right)n_2\omega }\right)(14)$

and

$H_1\left(G_2\vee k_1\right)=\frac{1}{2\left(n_2+1\right)}\left({\displaystyle \sum _{j=2}^{n_2}}\frac{1}{N\omega +{\mu }_j}+\frac{1}{\left(n_2+1\right)N\omega }\right).(15)$

Equation 11 is then obtained by combining Equations 2, 7, 14, 15.

Remark 2. Setting $N=1$ in Equation 11, we have

$H_1\left(G_1◦G_2\right)=H_1\left(G_1\right)+H_1\left(G_2\vee k_1\right)+\frac{\left(n_1-1\right)n_2}{2n_1{\left(n_2+1\right)}^2\omega },(16)$

which is consistent with the result of [46] (see Theorem 3 of [46] for details).

In the proof of Theorem 2, the first-order coherence of the join graphs $G_1\vee k_1$ and $G_2\vee k_1$ is also derived, as presented in Equations 14, 15, respectively. From Equation 14, $H_1(G_1\vee k_1)$ generally increases with the increase of $H_1(G_1)$. The assertion holds true also for $H_1(G_2\vee k_1)$ with $H_1(G_2)$. Therefore, from Equation 11, lower $H_1(G_1)$ or $H_1(G_2)$ generally indicates lower $H_1(G_1◦G_2)$. Furthermore, for a fixed $\omega$, we have $H_1(G_1◦G_2)\to (1+\frac{1}{n_1})H_1(G_1\vee k_1)+\frac{1}{2n_2(n_1+1)\omega }$ as $N\to \infty$. On the other hand, given a constant $N$, $H_1(G_1◦G_2)\to \frac{1}{N}{H}_1(G_1)$ as $\omega \to \infty$. The above analysis leads to the following remark.

Remark 3. From Equation 11, Theorem 2 shows, for fixed values of $n_1$, $n_2$, $N$ and $\omega$, lower $H_1(G_1)$ or $H_1(G_2)$ generally leads to lower $H_1(G_1◦G_2)$. In other words, higher first-order coherence of the factor network $G_1$ or $G_2$ usually implies higher first-order coherence of the $N$-duplication weighted corona network $G_1◦G_2$. Especially, in the situation of large duplication $N$ or weight $\omega$, the first-order coherence of $G_1◦G_2$ is mainly determined by the factor network $G_1$.

3.2 Second-order coherence of the $N$-duplication weighted corona network

This subsection investigates the second-order coherence of the $N$-duplication weighted corona networks.

Theorem 3. Let $\sigma (G_1)=\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{n_1}|0={\lambda }_1<{\lambda }_2\le \cdots \le {\lambda }_{n_1}\}$ and $\sigma (G_2)=\{{\mu }_1,{\mu }_2,\dots ,{\mu }_{n_2}|0={\mu }_1<{\mu }_2\le \cdots \le {\mu }_{n_2}\}$ be the Laplacian spectrum sets of $G_1$ and $G_2$, respectively. Then, the second-order coherence of the $N$-duplication weighted corona network $G_1◦G_2$ can be described as follows:

$\begin{array}{cc}{H}_2\left(G_1◦G_2\right)=\hfill & \frac{n_2}{n_1\left(n_2+N\right)N^2\omega }{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i}+\frac{n_2+N}{2n_1{N}^2}{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i^2}\hfill \\ \hfill & +\frac{N-1}{2n_1\left(n_2+N\right)}{\displaystyle \sum _{i=1}^{n_1}}\frac{1}{{\left(n_2\omega +{\lambda }_i\right)}^2}\hfill \\ \hfill & +\frac{1}{2\left(n_2+N\right)}{\displaystyle \sum _{j=2}^{n_2}}\frac{1}{{\left(N\omega +{\mu }_j\right)}^2}+\frac{1}{2n_1{\left(n_2+N\right)}^3{\omega }^2}\hfill \\ \hfill & +\frac{n_1-1}{2n_1\left(n_2+N\right)N^2{\omega }^2}.\hfill \end{array}(17)$

Proof. Based on Equation 4, to evaluate the second-order network coherence of $G_1◦G_2$, we need to obtain the sum of squared reciprocals of all nonzero Laplacian eigenvalues. The analysis of the Laplacian spectrum of $G_1◦G_2$ is presented in Theorem 1. For Equation 8, the last factor of the Laplacian characteristic polynomial Equation 6, the constant term $a_0$, and the coefficient of the linear term $a_1$ are given in Equation 9. Let $a_2$ denote the coefficient of the quadratic term of Equation 8. We have

$a_2=\prod _{i=2}^{n_1}\left(N\omega {\lambda }_i\right)\left[{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{N\omega {\lambda }_i}+{\displaystyle \sum _{i=2}^{n_1-1}}{\displaystyle \sum _{j=i+1}^{n_1}}\frac{\left(N\omega +n_2\omega +{\lambda }_i\right)\left(N\omega +n_2\omega +{\lambda }_j\right)}{N^2{\omega }^2{\lambda }_i{\lambda }_j}\right]$

and

${\left(\frac{a_1}{a_0}\right)}^2-2\frac{a_2}{a_0}=\frac{2n_2}{N^2\omega }{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i}+\frac{{\left(N+n_2\right)}^2}{N^2}{\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\lambda }_i^2}+\frac{n_1-1}{N^2{\omega }^2}.(18)$

The result of the theorem is then deduced by combining the analysis of the Laplacian spectrum of $G_1◦G_2$, Lemma 1, and Equations 4, 18.

From Equation 17, a conclusion similar to Remark 1 can be drawn for the second-order network coherence in $N$-duplication weighted corona networks.

Remark 4. Theorem 3 implies that $H_2(G_1◦G_2)$ decreases as the duplication $N$ or the weight factor $\omega$ increases. Therefore, similar to the case of first-order coherence, second-order noisy corona networks $G_1◦G_2$ with larger duplication $N$ or weight factor $\omega$ can be considered to be more robust to noise than networks with smaller duplication or weight factor values.

From Remark 4, the notable topological property of high coherence of corona networks $G_1◦G_2$ with large duplication $N$ remains valid for $N$-duplication weighted corona networks with second-order dynamics. As in the case of the first-order coherence, the relationship between the second-order coherence of the corona network $G_1◦G_2$ and that of the factor networks $G_1$ and $G_2$ is also explored.

Theorem 4. Let $k_1$ denote the trivial graph, $G_1\vee k_1$ be the join graph of $G_1$ and $k_1$ with the join-weight $n_2\omega$, and $G_2\vee k_1$ be the join graph of $G_2$ and $k_1$ with the join-weight $N\omega$. Then, the second-order coherence of the $N$-duplication weighted corona network $G_1◦G_2$ can be described as follows:

$\begin{array}{cc}{H}_2\left(G_1◦G_2\right)=\hfill & \frac{2n_2}{\left(n_2+N\right)N^2\omega }{H}_1\left(G_1\right)+\frac{n_2+N}{N^2}{H}_2\left(G_1\right)\hfill \\ \hfill & +\frac{\left(n_1+1\right)\left(N-1\right)}{n_1\left(n_2+N\right)}{H}_2\left(G_1\vee k_1\right)+\frac{n_2+1}{n_2+N}{H}_2\left(G_2\vee k_1\right)\hfill \\ \hfill & +\frac{1}{2\left(n_2+N\right){\omega }^2}\left[\frac{n_1-1}{n_1{N}^2}+\frac{1}{n_1{\left(n_2+N\right)}^2}+\frac{\left(n_1+2\right)\left(N-1\right)}{n_2^2{\left(n_1+1\right)}^2}\right.\hfill \\ \hfill & \left.-\frac{1}{{\left(n_2+1\right)}^2{N}^2}\right].\hfill \end{array}(19)$

Proof. The Laplacian eigenvalues of $G_1\vee k_1$ and $G_2\vee k_1$ are given in Equations 12, 13, respectively. From Equation 4, one obtains

$H_2\left(G_1\vee k_1\right)=\frac{1}{2\left(n_1+1\right)}\left({\displaystyle \sum _{i=2}^{n_1}}\frac{1}{{\left(n_2\omega +{\lambda }_i\right)}^2}+\frac{1}{{\left(n_1+1\right)}^2{n}_2^2{\omega }^2}\right)(20)$

and

$H_2\left(G_2\vee k_1\right)=\frac{1}{2\left(n_2+1\right)}\left({\displaystyle \sum _{j=2}^{n_2}}\frac{1}{{\left(N\omega +{\mu }_j\right)}^2}+\frac{1}{{\left(n_2+1\right)}^2{N}^2{\omega }^2}\right).(21)$

The theorem is then proved by combining the analysis of the Laplacian spectrum of $G_1◦G_2$, Lemma 1, and Equations 17, 20, 21.

Remark 5. Setting $N=1$ in Equation 19, the second-order network coherence of the 1-duplication corona network (or simply the corona network) can be expressed as follows:

$\begin{array}{ll}\hfill H_2\left(G_1◦G_2\right)& =\frac{2n_2}{\left(n_2+1\right)\omega }{H}_1\left(G_1\right)+\left(n_2+1\right)H_2\left(G_1\right)+H_2\left(G_2\vee k_1\right)\hfill \\ \hfill & +\frac{n_2\left(n_1-1\right)\left(n_2+2\right)}{2n_1{\left(n_2+1\right)}^3{\omega }^2}.\hfill \end{array}(22)$

In the proof of Theorem 4, the second-order coherence for the join graphs $G_1\vee k_1$ and $G_2\vee k_1$ is derived, as shown in Equations 20, 21, respectively. From Equation 20, it can be seen that $H_2(G_1\vee k_1)$ generally increases with the increase of $H_2(G_1)$. The assertion holds also true for the relationship between $H_2(G_2\vee k_1)$ and $H_2(G_2)$. Therefore, from Equation 19, lower $H_1(G_1)$, $H_2(G_1)$, or $H_2(G_2)$ generally leads to lower $H_2(G_1◦G_2)$. Furthermore, for a fixed $\omega$, $H_2(G_1◦G_2)\to (1+\frac{1}{n_1})H_2(G_1\vee k_1)+\frac{n_1+2}{2n_2^2{(n_1+1)}^2{\omega }^2}$ as $N\to \infty$. On the other hand, given a constant $N$, $H_2(G_1◦G_2)\to \frac{n_2+N}{N^2}{H}_2(G_1)$ as $\omega \to \infty$. The above analysis leads to the following remark.

Remark 6. Similar to the first-order noisy consensus algorithms, for fixed values of $n_1$, $n_2$, $N$, and $\omega$, the higher second-order coherence of the factor network $G_1$ or $G_2$ generally implies higher second-order coherence of the corona network $G_1◦G_2$. Especially, in the situation of large duplication $N$ or weight $\omega$ values, the second-order coherence of $G_1◦G_2$ is mainly determined by the factor network $G_1$.

4 Examples and simulations

This section verifies the theoretical results of Section 3 with numerical examples.

Example 1: Consider the network coherence of the $N$-duplication weighted corona network $G_1◦G_2$, where $G_1$ and $G_2$ are complete graphs of orders 5 and 2, respectively. Examples of 1-duplication and 2-duplication corona networks are shown in Figure 1.

The Laplacian eigenvalues of $G_1$ and $G_2$ are $0,\underset{4}{\underbrace{5,\dots ,5}}$ and $\mathrm{0,2}$, respectively. The first- and second-order coherence of the $N$-duplication weighted corona network $G_1◦G_2$ can be derived from Equations 7, 17, respectively. Especially, setting $\omega =1$, $H_1(G_1◦G_2)\to \frac{1}{2n_1}{\sum }_{i=1}^{n_1}\frac{1}{n_2+{\lambda }_i}\approx 0.1071$, and $H_2(G_1◦G_2)\to \frac{1}{2n_1}{\sum }_{i=1}^{n_1}\frac{1}{{(n_2+{\lambda }_i)}^2}\approx 0.0332$, as $N\to \infty$; setting $N=1$, $H_1(G_1◦G_2)\to \frac{1}{2n_1}{\sum }_{i=2}^{n_1}\frac{1}{{\lambda }_i}=H_1(G_1)=0.08$ and $H_2(G_1◦G_2)\to \frac{n_2+1}{2n_1}{\sum }_{i=2}^{n_1}\frac{1}{{({\lambda }_i)}^2}=(n_2+1)H_2(G_1)=0.048$ as $\omega \to \infty$.

Figure 2 shows the asymptotic trend of $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ with the increasing duplication $N$ and weight factor $\omega$, respectively. Furthermore, we can see the steep decline of $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ occurring at the small values of $N$ and $\omega$.

Figure 2

Figure 2. (a) $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ versus duplication $N$; (b) $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ versus weight factor $\omega$.

The dependence of $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ on duplication $N$ and weight $\omega$ is depicted in Figure 3. It can be seen from the hook face that both $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ generally decrease as $\omega$ or $N$ increases. Accordingly, a corona network with large duplication $N$ and weight factor $\omega$ can be considered to be more robust to noise than networks with small $N$ and $\omega$.

Figure 3

Figure 3. (a) $H_1(G_1◦G_2)$ with various duplication $N$ and weight factor $\omega$ values; (b) $H_2(G_1◦G_2)$ with various duplication $N$ and weight factor $\omega$ values.

Example 2: In this example, the relationship between the coherence of the $N$-duplication weighted corona network $G_1◦G_2$ and that of the factor networks $G_1$ and $G_2$ is explored. To this aim, two different cases are considered.

In case (I), the $N$-duplication weighted corona networks $G_1◦G_2$ are composed of the same factor graph $G_2$ (complete graph of order 5) but different $G_1$ (complete, cycle, and star graphs, all with eight vertices). The first- and second-order coherence of the three different $G_1$ is $H_1(G_c)=0.0547$, $H_1(G_{cy})=0.2613$, $H_1(G_s)=0.3828$, $H_2(G_c)=0.0068$, $H_2(G_{cy})=0.2604$, and $H_2(G_s)=0.3760$, where the subscripts $c$, $cy$, and $s$ stand for the complete, cycle, and star graph, respectively. The results of case (I) are shown in Figure 4.

Figure 4

Figure 4. $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ of the $N$-duplication weighted corona networks $G_1◦G_2$ composed of the same factor graph $G_2$ but different $G_1$, where (a, b) are versus duplication $N$ and (c, d) are versus weight $\omega$. The different factor graphs $G_1$ are a complete graph (C), a cycle graph (Cy), and a star graph (S), respectively.

In case (II), the $N$-duplication weighted corona networks $G_1◦G_2$ are composed of the same factor graph $G_1$ (complete graph of order 8) but different $G_2$ (complete, cycle, and star graphs, all with five vertices). The first- and second-order coherence of the three different $G_2$ is $H_1(G_c)=0.0800$, $H_1(G_{cy})=0.1632$, $H_1(G_s)=0.3200$, $H_2(G_c)=0.0160$, $H_2(G_{cy})=0.0808$, and $H_2(G_s)=0.3040$, respectively. The results of case (II) are shown in Figure 5.

Figure 5

Figure 5. $H_1(G_1◦G_2)$ and $H_2(G_1◦G_2)$ of the $N$-duplication weighted corona networks $G_1◦G_2$ composed of the same factor graph $G_1$ but different $G_2$, where (a, b) are versus duplication $N$ and (c, d) are versus weight $\omega$. The different factor graphs $G_2$ are a complete graph (C), a cycle graph (Cy), and a star graph (S), respectively.

The influence of the factor networks $G_1$ and $G_2$ on the coherence of the corona network $G_1◦G_2$ is illustrated in Figures 4, 5, respectively. As shown in Figure 4, in the case of the same factor graph $G_2$, a smaller value of $H_1(G_1)$ (or $H_2(G_1)$) also generally leads to a smaller value of $H_1(G_1◦G_2)$ (or $H_2(G_1◦G_2)$). In other words, for the same $G_2$, the $N$-duplication weighted corona network $G_1◦G_2$ will generally have higher network coherence when the factor network $G_1$ has higher network coherence. From Figure 5, it can be seen that the above assertion also holds true for the influence of $G_2$ on the network coherence of $G_1◦G_2$. Moreover, compared with the results of Figure 5, the values of $H_1(G_1◦G_2)$ (or $H_2(G_1◦G_2)$) in Figure 4 show a more notable difference, which indicates that the factor network $G_1$ plays a more important role than $G_2$ in the network coherence of the $N$-duplication weighted corona network $G_1◦G_2$.

5 Conclusion

In this paper, coherence problems in $N$-duplication weighted corona networks with first- or second-order dynamics are addressed. As a special case with N =1, the network coherence in the classic corona network is also investigated (see Equations 16, 22). For both first- and second-order consensus problems, explicit expressions of the network coherence are derived and presented in two different ways. In one way, the network coherence is expressed in terms of the Laplacian spectra of the factor networks $G_1$ and $G_2$, the weight factor $\omega$ of edges connecting $G_1$ and $G_2$, and the duplication $N$. Based on this kind of expression, it is found that corona networks with large duplication $N$ or weight $\omega$ usually have high network coherence and can be considered to be more robust to noise. High coherence or strong robustness of the corona network $G_1◦G_2$ with large duplication $N$ (thereby large network size) can be regarded as a special and notable topological property of the $N$-duplication weighted corona network. In another way, the coherence of the corona network $G_1◦G_2$ is expressed in terms of that of the factor networks $G_1$ and $G_2$. Little work has been done from this perspective on the study of the consensus problems in product networks, and it deserves further research. Based on this kind of expression, the influence of the factor networks $G_1$ and $G_2$ on the network coherence of the corona network $G_1◦G_2$ is investigated. The results show that higher coherence of $G_1$ or $G_2$ usually also leads to higher coherence of the corona network $G_1◦G_2$. Especially, in the situation of large duplication $N$ or weight $\omega$, $G_1$ plays a more important role than $G_2$ in the network coherence of the $N$-duplication weighted corona network $G_1◦G_2$.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.

Author contributions

CL: Writing – original draft and Writing – review and editing.

Funding

The author(s) declare that no financial support was received for the research and/or publication of this article.

Conflict of interest

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

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Abbreviations

$◦$ $N$-duplication corona product; $\vee$ join operation; $\bigcup$ union of sets; $\otimes$ Kronecker product; $R$ real number field; $I_k$ $k$-dimensional identity matrix; $1_k$ $k$-dimensional vector with all elements being 1.

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