Abstract
The consensus of complex networks has attracted the attention of many scholars. The graph operation is a common method to construct complex networks, which is helpful in studying the consensus of complex networks. Based on the corona networks G1◦G2, this study gives different weights to the edges of G1◦G2 to obtain the weighted corona networks and studies the consensus of . The consensus of the networks can be measured by coherence. First, the Laplacian polynomial of is derived by using the properties of an orthogonal matrix. Second, the relationship between the first-order coherence of and G1 is deduced by using the relevant properties of the determinant and the conclusion of polynomial coefficients and the principal minors of the matrix. Third, the join operation is introduced to further simplify the analytical formula of network coherence. Finally, a specific network example is used to verify the validity of the conclusion.
1 Introduction
With the development of network science, the research of complex networks has been extended to many fields, such as technical networks and transportation networks. Nowadays, the relevant theoretical knowledge of complex networks has been widely used in physics, computer science, life science, and other fields, such as consensus [1–5], resistance distance and Kirchhoff index [6], robustness [7, 8], and network synchronization [9, 10].
As a method of constructing networks, the graph operation can be used to construct more complex networks. The common graph operations include corona operation, edge corona operation, and join operation. In recent years, graph operations have attracted extensive attention of scholars. Y. Shang used the edge corona product to construct a simplicial network and, based on the degree of network vertices, studied the recently widely concerned Sombor index [11]. J. Liu presented a kind of weighted edge corona networks and obtained the Laplacian and signless Laplacian spectra of the weighted edge corona networks, and a specific application example is given by calculating the number of spanning trees and the Kirchhoff index [12]. M. Dai used the eigenvector method to obtain the generalized adjacency and Laplacian spectra of the special weighted corona networks [13]. We considered the weighted corona networks that are more realistic as the research object to study the consensus of networks.
The consensus of the networks is the key to solve cooperative control among nodes in complex networks. The consensus of complex networks means that network nodes reach the same level in a certain state with the change of time. For example, the direction of unmanned aerial vehicle formation is consistent during flight. The research on the consensus of special networks has achieved many good results. E. Mackin took the network of networks as the object, investigated how to connect the subgraph to achieve the optimal consensus of the networks, and used a specific example to illustrate it [14]. T. Hu defined three types of tree models with the given parameters and obtained the leaderless and leader–follower coherence of three types of network models. The study found that the leader–follower coherence was weaker than the leaderless coherence [15]. J. Wang analyzed the consensus of three different types of weighted duplex networks and compared the consensus of the three types of networks [16]. J. Chen showed the consensus of a class of special topological networks and obtained the relationships between the network consensus and parameters [17]. X. Wang used the property of the determinant to calculate the Laplacian polynomial of 5-rose graphs and investigated the consensus of 5-rose graphs by using the relationships between polynomial coefficients and eigenvalues [18].
Compared with the aforementioned literature, the innovation of this study is as follows. This study defines the weighted corona network model based on the unweighted corona networks. We used the properties of the orthogonal matrix to transform a high-order determinant into a low-order determinant and deduced the Laplacian polynomial of the weighted corona networks. Finally, the specific analytical formula of the first-order coherence of the weighted corona networks is obtained according to the relationship between the coefficient and the principal minor, which provided a theoretical basis for studying the coherence of the arbitrary weighted corona network.
This study is arranged as follows. Section 2 introduces the preliminaries. The characteristic polynomial of is given in Section 3. Section 4 obtains the first-order coherence of . In Section 5, the specific application example is shown. Section 6 gives the final conclusion.
2 Preliminaries
2.1 Definitions of the weighted graph operations
The topology of networks is the key to study the consensus of complex networks. It is a common method to construct complex networks by using graph operations. Next, we introduced two graph operations.
Definition 1 [19, 20]: Let G1 and G2 denote the two graphs with n1 and n2 vertices, respectively. The corona of G1 and G2 is described as the graph G1◦G2 obtained by taking one copy of G1 and n1 copies of G2 and then joining the ith vertex of G1 to every vertex in the ith copy of G2 (i = 1, 2, 3, … , n1).
The weighted corona graphs mean that on the basis of G1◦G2, the edges of G1 and G2 have weights r1 and r2, respectively. The edges between G1 and G2 have weight r3. is shown in Figure 1.
Definition 2 [21]: Let the join of two disjoint graphs G1 and G2 be G1 ∨ G2, the vertex set of G1 ∨ G2 be V (G1 ∨ G2) = V (G1) ∪ V (G2), and the edge set of G1 ∨ G2 be E (G1 ∨ G2) = E (G1) ∪ E (G2) ∪ (xy) (x ∈ V (G1), y ∈ V (G2)).
FIGURE 1
The weighted join graphs mean that on the basis of G1 ∨ G2, the edges of G1 and G2 have weights r1 and r2, respectively. The edges between G1 and G2 have weight r3.
2.2 Network coherence
The network model of the system with noise is defined as follows [4]:where L = D − A is the Laplacian matrix, D = diag (d1, d2, … , ds) denotes the degree matrix, di (i = 1, 2, … , s) is the degree of the ith node of the network. is the adjacency matrix, where , and x(t) denotes the dynamic variable of the network nodes. χ(t) indicates that all nodes in the network are affected by Gaussian white noise at time t. The Gaussian white noise means that the instantaneous value of noise obeys Gaussian distribution. It is an ideal model for noise analysis.
Under the influence of noise, it is difficult for all nodes in the network to converge in a certain state. In order to describe the consensus of the networks, the concept of first-order coherence is introduced. It is defined as follows [4]:
The output of system (1) is denoted as follows:where K is the projection operator, , and 1 is the s-vector of all nodes. H(1) is given by the H2 norm of the systems defined in Eqs 1 and 3.
The research shows that H(1) is closely related to the non-zero eigenvalues λi (i = 2, 3, … , s) of the Laplacian matrix L [18],
The Kirchhoff index (Kf) is also closely related to the non-zero eigenvalues of the Laplacian matrix L, .
3 Laplacian polynomial of
In this section, the Laplacian polynomial of is obtained by using the properties of the orthogonal matrix.
Lemma 1: Let the Laplacian eigenvalues of G be 0 = v1 < v2 ≤ v3 ≤ ⋯ ≤ vn; there is an orthogonal matrix . Then, , where pi (i = 1, 2, … , n) is the sum of the ith column of the matrix P.
Proof: the Laplacian eigenvalues of G are 0 = v1 < v2 ≤ v3 ≤ ⋯ ≤ vn; then, there is an orthogonal matrix . It is obvious that is the unit eigenvector of the eigenvalue . Let be the unit eigenvector corresponding to the eigenvalue vi (i = 2, 3, … , n); from the orthogonality of the eigenvectors, we have (1, … , 1)ζi = 0 (i = 2, 3, … , n) and p2 = p3 = ⋯ = pn = 0.Theorem 1: let the number of vertices of G1 and G2 be n1 and n2, respectively; the Laplacian eigenvalues of G1 and G2 are . Then, the Laplacian polynomial of isProof: let Ai (i = 1, 2) be the adjacency matrices of G1 and G2; Di (i = 1, 2) denotes the degree matrices of G1 and G2; is the identity matrix of order n1; represents all 1 column vector of dimension n2; and 0m×n represents a zero matrix.
The adjacency matrix of is
The degree matrix of is
The Laplacian matrix of is
Let the Laplacian eigenvalues of G1 and G2 be
Then, there are orthogonal matrices , and
Because similar matrices have the same characteristic polynomials, the Laplacian polynomial of is
In order to find a specific expression for Φ(λ), we further investigated ,where is the sum of the jth column of the orthogonal matrix N. By Lemma 1 and Eq. 10, we obtained
By Eq. 8,
By formulas (9), (11), 12) and (13), the row and column of element λ − r3 − r2μi (i = 2, 3, … , n2) are all 0 except itself. We expanded them according to the row; then,
4 First-order coherence of
In this section, according to theorem 1, the first-order coherence of is calculated by using the relationship between characteristic polynomial coefficients and the principal minors the of matrix.
Theorem 2: the first-order coherence of can be described as follows:Proof: according to theorem 1, the Laplacian polynomial of is
However, the Laplacian matrix of must contain a zero eigenvalue, and r3 + r2μi ≠ 0 (i = 2, 3, … , n2).
Therefore, let be the eigenvalues of ψ(λ),
By [16], we have
Just for the sake of calculation, let B*(i) and B*(i, j) be the submatrices of matrix B by removing the ith row, and ith and jth rows. B†(i) and B†(i, j) are the submatrices of matrix B by removing the ith column, and ith and jth columns.
. Here, |C(i)| represents the (2n1 − 1)-order principal minors of matrix C by removing the ith row and column, and |C (i, j)| represents the (2n1 − 2)-order principal formula of matrix C by removing the ith and jth rows, and ith and jth columns. Ei is the diagonal matrix, where the ith element is r3n2 + r3, and the remaining elements are all 0.
First, we calculated a1, and we obtained it from algebra,
When 1 ≤ i ≤ n1, the (n1 − 1 + i)-th row of |C(i)| is all 0; then, ,where t (G1) is the number of spanning trees of G1.
Second, we calculated a2,
If 1 ≤ i < j ≤ n1, the (n1 − 2 + i)-th and (n1 − 2 + j)-th rows of |C (i, j)| are all 0; then, .
If 1 ≤ i ≤ n1, j > n1, j ≠ n1 + i and the (n1 − 1 + i)-th row of |C (i, j)| is all 0, then .
By Eq. 16,
where
.
By the properties of the determinant, we can obtain
By varying the determinant elementary column, we obtain
Because |r1L1| = 0, then
By formulas (15), (17), (18), and 19) and the literature [22], we obtain
From Eq. 5,
Since the form of theorem 2 is complicated, we further optimized the conclusion to obtain theorem 3.
Theorem 3Proof: the Laplacian eigenvalues of areBy Eq. 5,
Let K1 be the complete graph of order 1. The adjacency matrix of is
The degree matrix of is
The Laplacian matrix of is
Because of
The Laplacian eigenvalues of areThen,
By formulas (14), (21), and (22), we obtain
5 Actual example
Let be the complete bipartite graph of order n1 and be the cycle of order n2.
The Laplacian spectrum of is
The Laplacian spectrum of is
From Eq. 20,
From Eq. 21,
From Eq. 22,
Then,.
6 Conclusion
For the unweighted corona networks G1◦G2, the corresponding Laplacian spectra can be obtained by the eigenvector method, and we can use the relationship between eigenvalues and coherence to get the network coherence of G1◦G2. For the weighted corona networks , due to the different weights, it is difficult to obtain the Laplacian spectra, that is, it is not easy to obtain the network coherence of through the eigenvalue spectra. Based on the Laplacian matrix , the Laplacian characteristic polynomial of is calculated by using the properties of matrix diagonalization and orthogonal matrix. We further used the relationship between eigenvalues and characteristic polynomial coefficients to obtain the network coherence of and found the relationship between the network coherence of and G1. The Laplacian spectra of the weighted corona networks are not obtained in this study. We will conduct further research in future work. The research on the Sombor index and degree-related properties of simplicial networks is very meaningful, and we will try to obtain the Sombor index of the weighted corona networks.
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
Conceptualization, HG and WD; methodology, HG and JZ; software, WD; validation, XL and JZ; formal analysis, HG and WD; writing—original draft preparation, HG and WD; writing—review and editing, JZ; supervision, XL; and project administration, HG. All authors contributed to manuscript revision, read, and approved the submitted version.
Funding
This work was supported by the National Natural Science Foundation of China (no.61802316), the School level project of the Xinjiang institute of Light industry Technology (no. XJQG2022S16), the Scientific Research and Education Project of Xinjiang Institute of Engineering (2020xgy372302), the National innovation and entrepreneurship training program for College Students (no. 202110994006), and the project of Key Laboratory of New Energy and Materials Research of Xinjiang Institute of Engineering.
Acknowledgments
The authors express their sincere gratitude to the people who gave valuable comments.
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.
Publisher’s note
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Summary
Keywords
consensus, weighted, corona operation, join operation, first-order coherence
Citation
Du W, Zhu J, Gao H and Li X (2022) Consensus analysis of the weighted corona networks. Front. Phys. 10:948247. doi: 10.3389/fphy.2022.948247
Received
19 May 2022
Accepted
15 July 2022
Published
19 August 2022
Volume
10 - 2022
Edited by
Mahdi Jalili, RMIT University, Australia
Reviewed by
Weigang Sun, Hangzhou Dianzi University, China
Yilun Shang, Northumbria University, United Kingdom
Yongqing Wu, Liaoning Technical University, China
Updates
Copyright
© 2022 Du, Zhu, Gao and Li.
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: Haiping Gao, yqh@xjie.edu.cn
This article was submitted to Statistical and Computational Physics, a section of the journal Frontiers in Physics
Disclaimer
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.