Abstract
This paper studies first- and second-order coherence problems for -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 , the weight , and the factor networks and on the network coherence of the corona network is investigated. Some noteworthy topological properties of the -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 -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 -duplication corona product network was investigated in [50]. An example of this -duplication corona network is shown in Figure 1, where and are complete graphs of order 5 and 2, respectively.
FIGURE 1
Illustration of -duplication corona graphs composed of and , 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 -duplication corona product network, which includes the basic corona network as a special case with the duplication . The intricate topological configurations of the -duplication corona product network not only compounds analytical challenges in coherence studies but also raises new research questions. What influence will the duplication or the weight factor have on the network coherence? What is the relationship between the coherence of the corona network and that of its factor networks and ? 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 -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 -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 and the weight on the network coherence are explored. It is found that corona networks with larger duplication or higher weight 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 can be regarded as a special topological characteristic of the -duplication weighted corona network. Finally, based on the obtained formulas, the relationship between the coherence of the composite corona network and that of the factor networks and 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 or usually also leads to higher coherence of the corona network . Especially in the situation of large duplication or high weight , the network coherence of the corona network is mainly determined by the factor network . 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 -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 -duplication weighted corona networks. The influence of the duplication , the weight , and the factor networks and on the network coherence of the corona network 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 -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 , where is the set of nodes, is the set of edges, and is the weighted adjacency matrix. The Laplacian matrix is defined as , where is the degree matrix. The network has consensus dynamics modeled by the stochastic differential equationwhere is the state of the system, is the Laplacian matrix, and is an -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,It has been shown [6–9] that can be completely determined by the eigenvalues of the Laplacian matrix ,where 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 and a velocity vector . The second-order consensus dynamics subject to noise are described bywhere 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 only and denoted by the mean (over all nodes) and the steady-state variance of the deviation from the average of ,This value is also completely determined by the eigenvalues of the Laplacian matrix [8, 9, 15],where are the Laplacian eigenvalues.
Note that networks with smaller steady-state variance or 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] Letbe a real polynomial of degreewith. It showsandwhereare the roots of.
2.4 -duplication weighted corona networks
As an extension of the classic corona network, an -duplication weighted corona network was recently introduced in the literature [49, 50].
Definition 1[49, 50] Letandbe two finite, simple, nonempty, and vertex-disjoint graphs withandvertices, respectively. The-duplication weighted corona productis generated by takingcopies ofandcopies ofand then joining theth vertex of eachto every vertex in theth copy of. All the weights of the newly added edges betweenandare the same and positive, denoted as.
Let and be the Laplacian matrices of and respectively, then the Laplacian matrix of the -duplication weighted corona product is
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 graphsand, denoted by, is the graph with vertex setand the edge set.represents the added edge joiningand. Each linkhas the same and positive weight, called the join-weight of.
In [49, 50], the characteristic polynomial of the -duplication weighted corona graph was presented based on the eigenvalues of the factor graphs and .
Lemma 2[49, 50] Letandbe the Laplacian spectrum sets ofand, respectively. Then, the Laplacian characteristic polynomial of the-duplication weighted corona productwith the Laplacian matrixEquation 5is
3 Main results
This section studies the first- and second-order coherence problems for -duplication weighted corona networks, where vertices are subject to white noise. Note that the graphs , and the -duplication weighted corona network used in the sequel are all as defined in Def. 1.
3.1 First-order coherence of the -duplication weighted corona network
Letandbe the Laplacian spectrum sets ofand, respectively. Then the first-order coherence of the-duplication weighted corona networkcan be described as follows:
Proof. By Def. 1, the -duplication weighted corona network contains vertices. From Lem. 2, the Laplacian eigenvalues of are the roots of . Then, the Laplacian spectrum of consists of
(i) 0;
(ii) ;
(iii) , repeated times for ;
(iv) , repeated times for ;
(v) Two roots of the equationFrom Equation 2, the first-order network coherence of 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:Let and denote the constant term and the coefficient of the linear term of Equation 8, respectively. Then,and
Combining the Laplacian spectrum of as shown in (i)-(v), Lemma 1, and Equations 2, 10, one obtains
The proof is completed.
Remark 1FromEquation 7, Theorem 1 implies thatdecreases as the duplicationor the weight factorincreases. Accordingly, corona networkswith larger duplicationor weight factorhave 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 with a large duplication 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 with a large duplication (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 -duplication weighted corona network can be considered a graph with good robustness to external disturbances, which provides new insights into its practical applications.
In Equation 7, is characterized by the Laplacian eigenvalues of the associated matrices and , the weight , and the duplication . To further explore the relationship between the first-order coherence of the composite network and that of the factor networks and , we derive another analytical formula for .
Lemma 3Letbe a simple graph withvertices and Laplacian eigenvalues. There is an orthogonal matrix, such that. Moreover,, whereis the sum of theth column of the matrix.
Proof. For the Laplacian matrix , it is obvious that is the unit eigenvector associated with eigenvalue . Let be an eigenvector of associated with eigenvalue . Then, ; that is to say, the sum of all the entries of is 0. The conclusion of the lemma follows from the orthogonal decomposition theorem.
Letdenote the complete graph of order 1 (i.e., the trivial graph),be the join graph ofandwith the join-wight, andbe the join graph ofandwith the join-wight. Then, the first-order coherence of the-duplication weighted corona networkcan be described as follows:
Proof. Let and denote the adjacency and Laplacian matrices of , respectively. The block form of the adjacency matrix of iswhere represents the all-ones column vector of dimension . The Laplacian matrix of is
Suppose that is the spectrum set of . Then, there is an orthogonal matrix such that . In addition,From Lemma 3, the characteristic polynomial of isTherefore, the Laplacian eigenvalues of with the join-weight areSimilarly, the Laplacian eigenvalues of with the join-weight areMoreover, we haveandEquation 11 is then obtained by combining Equations 2, 7, 14, 15.
Remark 2SettinginEquation 11, we havewhich 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 and is also derived, as presented in Equations 14, 15, respectively. From Equation 14, generally increases with the increase of . The assertion holds true also for with . Therefore, from Equation 11, lower or generally indicates lower . Furthermore, for a fixed , we have as . On the other hand, given a constant , as . The above analysis leads to the following remark.
Remark 3FromEquation 11,Theorem 2shows, for fixed values of,,and, lowerorgenerally leads to lower. In other words, higher first-order coherence of the factor networkorusually implies higher first-order coherence of the-duplication weighted corona network. Especially, in the situation of large duplicationor weight, the first-order coherence ofis mainly determined by the factor network.
3.2 Second-order coherence of the -duplication weighted corona network
This subsection investigates the second-order coherence of the -duplication weighted corona networks.
Letandbe the Laplacian spectrum sets ofand, respectively. Then, the second-order coherence of the-duplication weighted corona networkcan be described as follows:
Proof. Based on Equation 4, to evaluate the second-order network coherence of , we need to obtain the sum of squared reciprocals of all nonzero Laplacian eigenvalues. The analysis of the Laplacian spectrum of is presented in Theorem 1. For Equation 8, the last factor of the Laplacian characteristic polynomial Equation 6, the constant term , and the coefficient of the linear term are given in Equation 9. Let denote the coefficient of the quadratic term of Equation 8. We haveand
The result of the theorem is then deduced by combining the analysis of the Laplacian spectrum of , 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 -duplication weighted corona networks.
Remark 4Theorem 3implies thatdecreases as the duplicationor the weight factorincreases. Therefore, similar to the case of first-order coherence, second-order noisy corona networkswith larger duplicationor weight factorcan 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 with large duplication remains valid for -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 and that of the factor networks and is also explored.
Letdenote the trivial graph,be the join graph ofandwith the join-weight, andbe the join graph ofandwith the join-weight. Then, the second-order coherence of the-duplication weighted corona networkcan be described as follows:
Proof. The Laplacian eigenvalues of and are given in Equations 12, 13, respectively. From Equation 4, one obtainsand
The theorem is then proved by combining the analysis of the Laplacian spectrum of , Lemma 1, and Equations 17, 20, 21.
Remark 5SettinginEquation 19, the second-order network coherence of the 1-duplication corona network (or simply the corona network) can be expressed as follows:
In the proof of Theorem 4, the second-order coherence for the join graphs and is derived, as shown in Equations 20, 21, respectively. From Equation 20, it can be seen that generally increases with the increase of . The assertion holds also true for the relationship between and . Therefore, from Equation 19, lower , , or generally leads to lower . Furthermore, for a fixed , as . On the other hand, given a constant , as . The above analysis leads to the following remark.
Remark 6Similar to the first-order noisy consensus algorithms, for fixed values of,,, and, the higher second-order coherence of the factor networkorgenerally implies higher second-order coherence of the corona network. Especially, in the situation of large duplicationor weightvalues, the second-order coherence ofis mainly determined by the factor network.
4 Examples and simulations
This section verifies the theoretical results of Section 3 with numerical examples.
Example 1: Consider the network coherence of the -duplication weighted corona network , where and 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 and are and , respectively. The first- and second-order coherence of the -duplication weighted corona network can be derived from Equations 7, 17, respectively. Especially, setting , , and , as ; setting , and as .
Figure 2 shows the asymptotic trend of and with the increasing duplication and weight factor , respectively. Furthermore, we can see the steep decline of and occurring at the small values of and .
FIGURE 2
(a) and versus duplication ; (b) and versus weight factor .
The dependence of and on duplication and weight is depicted in Figure 3. It can be seen from the hook face that both and generally decrease as or increases. Accordingly, a corona network with large duplication and weight factor can be considered to be more robust to noise than networks with small and .
FIGURE 3
(a) with various duplication and weight factor values; (b) with various duplication and weight factor values.
Example 2: In this example, the relationship between the coherence of the -duplication weighted corona network and that of the factor networks and is explored. To this aim, two different cases are considered.
In case (I), the -duplication weighted corona networks are composed of the same factor graph (complete graph of order 5) but different (complete, cycle, and star graphs, all with eight vertices). The first- and second-order coherence of the three different is , , , , , and , where the subscripts , , and stand for the complete, cycle, and star graph, respectively. The results of case (I) are shown in Figure 4.
FIGURE 4
and of the -duplication weighted corona networks composed of the same factor graph but different , where (a, b) are versus duplication and (c, d) are versus weight . The different factor graphs are a complete graph (C), a cycle graph (Cy), and a star graph (S), respectively.
In case (II), the -duplication weighted corona networks are composed of the same factor graph (complete graph of order 8) but different (complete, cycle, and star graphs, all with five vertices). The first- and second-order coherence of the three different is , , , , , and , respectively. The results of case (II) are shown in Figure 5.
FIGURE 5
and of the -duplication weighted corona networks composed of the same factor graph but different , where (a, b) are versus duplication and (c, d) are versus weight . The different factor graphs are a complete graph (C), a cycle graph (Cy), and a star graph (S), respectively.
The influence of the factor networks and on the coherence of the corona network is illustrated in Figures 4, 5, respectively. As shown in Figure 4, in the case of the same factor graph , a smaller value of (or ) also generally leads to a smaller value of (or ). In other words, for the same , the -duplication weighted corona network will generally have higher network coherence when the factor network has higher network coherence. From Figure 5, it can be seen that the above assertion also holds true for the influence of on the network coherence of . Moreover, compared with the results of Figure 5, the values of (or ) in Figure 4 show a more notable difference, which indicates that the factor network plays a more important role than in the network coherence of the -duplication weighted corona network .
5 Conclusion
In this paper, coherence problems in -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 and , the weight factor of edges connecting and , and the duplication . Based on this kind of expression, it is found that corona networks with large duplication or weight usually have high network coherence and can be considered to be more robust to noise. High coherence or strong robustness of the corona network with large duplication (thereby large network size) can be regarded as a special and notable topological property of the -duplication weighted corona network. In another way, the coherence of the corona network is expressed in terms of that of the factor networks and . 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 and on the network coherence of the corona network is investigated. The results show that higher coherence of or usually also leads to higher coherence of the corona network . Especially, in the situation of large duplication or weight , plays a more important role than in the network coherence of the -duplication weighted corona network .
Statements
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
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Abbreviations
-duplication corona product; join operation; union of sets; Kronecker product; real number field; -dimensional identity matrix; -dimensional vector with all elements being 1.
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Summary
Keywords
robustness, network coherence, N-duplication weighted corona network, Laplacian spectrum, join operation
Citation
Liu C (2025) First- and second-order network coherence in -duplication weighted corona networks. Front. Phys. 13:1562805. doi: 10.3389/fphy.2025.1562805
Received
18 January 2025
Accepted
13 May 2025
Published
20 June 2025
Volume
13 - 2025
Edited by
Adriano Tiribocchi, National Research Council (CNR), Italy
Reviewed by
Asad Ullah, Karakoram International University, Pakistan
Jian Zhu, China University of Mining and Technology, China
Updates
Copyright
© 2025 Liu.
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: Chao Liu, liu.chao@shufe.edu.cn
Disclaimer
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