CN112464414A - NW small-world network system synchronous analysis method based on spectrum moment - Google Patents

Abstract

The invention provides a spectral moment-based NW small world network system synchronization analysis method, which relates to the field of control and information technology. Based on the random matrix theory, the present invention studies the pinning and synchronization stability of complex networks from the perspective of the dynamic evolution of NW small-world networks. The conclusions obtained can better reveal the internal mechanism of the interaction between the structural evolution of the actual network system and the dynamic behavior, and then design for us. Provide theoretical and methodological guidance to meet the desired conditions (such as stability and robustness) of the actual network system, predict the behavior of the network system, and improve various performances of the actual network system through control means, and determine the factors that make the network achieve synchronization stability. conditions, with important economic and social significance.

Description

A Synchronization Analysis Method of NW Small World Network System Based on Spectral Moment

technical field

The invention relates to the technical field of control and information, in particular to a spectral moment-based synchronous analysis method for an NW small world network system.

Background technique

Synchronization of complex networks exists widely in nature and human society, and has attracted more and more attention in recent years. By controlling some nodes in the network, all nodes in the network tend to be in the same state, which is called the pinning synchronization problem of complex networks. Most of the existing research on the pinning and synchronization problem of complex networks assumes that the network is a deterministic network, and does not fully consider the randomness of topology evolution in actual network systems. The results of network synchronization analysis ignoring the randomness of such edge generation are greatly limited in practical applications. Therefore, it is necessary to develop a new network synchronization analysis method that can fully reflect the randomness of the evolution of the actual network topology and the randomness of edge generation.

SUMMARY OF THE INVENTION

Aiming at the deficiencies of the prior art, the present invention provides a spectral moment-based NW small-world network system synchronization analysis method. Based on random matrix theory, the present invention considers that the actual network system has the characteristics of random connection, which is helpful to understand the evolution law and dynamic mechanism of the complex network system structure in the real world, and then design the actual network system that meets the desired conditions for us. Provide theoretical and methodological guidance on networked systems, predicting the behavior of networked systems, and improving various performances of actual networked systems through control means.

The technical scheme adopted by the present invention is:

A spectral moment-based synchronization analysis method for NW small-world network systems, comprising the following steps:

Step 1: Construct a NW small world network with N nodes according to the NW small world network model construction algorithm;

是一个对称阵，N表示网络节点数目，

表示N×N维实数矩阵的集合，a_ij表示邻接矩阵元素，如果节点i与节点j之间有边相连，则a_ij＝a_ji＝1(i≠j)；否则a_ij＝a_ji＝0(i≠j)；d_i表示节点i的度，记为

D表示度矩阵，记为D＝diag(d_i)；

表示Laplacian(拉普拉斯)矩阵，l_ij表示Laplacian矩阵元素，记为L＝D-A；In the NW small world network, each node represents a coupled oscillator, and a network is formed by connecting edges between nodes, and each connecting edge represents the interaction relationship between the coupled oscillators. The network has a symmetrical connection structure. adjacency matrix

is a symmetric matrix, N represents the number of network nodes,

Represents a set of N×N-dimensional real number matrices, a _ij represents an adjacency matrix element, if there is an edge between node i and node j, then a _ij =a _ji =1 (i≠j); otherwise a _ij =a _ji = 0 (i≠j); d _i represents the degree of node i, denoted as

D represents the degree matrix, denoted as D=diag(d _i );

Represents the Laplacian matrix, and l _ij represents the elements of the Laplacian matrix, denoted as L=DA;

Step 2: Determine the dynamic equation expression of the nodes in the NW small world network system;

For a continuous-time dynamic network composed of N identical nodes, the dynamic equation of the ith node is expressed as:

表示第i个节点的状态变量，对时间t的一阶导数为

f(·)表示单个节点自身动力学函数；

为内耦合矩阵，表示各个节点状态变量之间的耦合关系；常数c>0表示全局耦合强度；in,

Represents the state variable of the ith node, and the first derivative with respect to time t is

f( ) represents the dynamic function of a single node itself;

is the internal coupling matrix, which represents the coupling relationship between the state variables of each node; the constant c>0 represents the global coupling strength;

Step 3: Analyze the synchronization stability of the NW small-world network with the main stability function method, and determine the synchronization and stability region of the NW small-world network system;

The synchronization stability of the NW small-world network, when x ₁ (t)=x ₂ (t)=...=x _N (t)=s(t), means that the NW small-world network achieves synchronization stability, where s(t) is called synchronous state; when pinning control is introduced, the dynamic equation of the controlled NW small-world network is expressed as:

in:

u _i (t)=-cf _i Γ(x _i (t)-s(t)),i=1,2,...,N (3)

同步态s(t)是单个节点系统

的一个解，满足

f _i represents the pinning control gain of the controlled node i, δ _i represents whether the i-th node is controlled or not, if pinning control is applied to node i, then δ _i =1, f _i >0; otherwise δ _i =fi ₌ 0 ; F=diag(f _i ) represents the pinning control gain matrix; l represents the number of pinning control nodes, denoted as

Synchronous state s(t) is a single node system

a solution that satisfies

I_n表示n×n维单位矩阵，

得：When the NW small-world network does not achieve synchronization and stability, there is an error between the state of each node and the synchronization state; the error vector is defined as ε _i (t), then x _i (t)=s(t)+ε _i (t); (2) For linearization and variable substitution, let

In represents an n× _n -dimensional identity matrix,

have to:

Among them, J _f (t) is the Jacobian matrix of f(x(t)) in synchronous state s(t); λ _i is the eigenvalue of matrix C=L+F; P is the equivalent transformation matrix of matrix C , the equivalent form of matrix C is denoted as Q=P ^-1 CP;

According to the main stable function method, the region where the maximum Lyapunov exponent of equation (4) is negative is the synchronous stable region of the NW small-world network system, denoted as SR={cλ _i |L _max (cλ _i )<0};

Step 4: Based on the spectral moment analysis method, according to the probability distribution of the network node degree, calculate the expected value of the first three-order spectral moment of the network matrix C after the introduction of the pinning control, and establish the relationship between the first three-order expected spectral moment and the network structure parameters, The network structure parameters include the number of network nodes N, the edge connection probability p, the pinning control gain f, and the number 1 of pinning control nodes;

The network matrix C is a real symmetric matrix, {λ _i , i=1,2,...,N} is denoted as the eigenvalue set of the matrix C, then the q-order spectral moment of the matrix C is expressed as:

At the initial moment, the network is a nearest-neighbor coupled network composed of N nodes, and the degree of each node is 2k; on the basis of this network, random edges are connected with probability p (to avoid self-loops and multiple edges), and an NW small world network is obtained. , and its node degree distribution obeys a Poisson distribution, that is,

Among them, d _i represents the degree of node i in the NW small world network, d represents the node degree value, and the parameter r=pN;

Assuming that the first l nodes in the NW small-world network are pinned and controlled, and all control gains are constant f, the expression of the expected value of the first third-order spectral moment of the matrix C is:

Step 5: use the expected value of the spectral moment obtained in step 4 to construct the eigenvalue spectrum of the piecewise linear function fitting matrix C;

According to the expected value of the first three-order spectral moments obtained in step 4, a triangular distribution function is used to construct the distribution of the piecewise function linear fitting characteristic spectrum;

The expected values of the first third-order spectral moments are denoted as m ₁ , m ₂ , and m ₃ ; in order to estimate the three eigenvalues of the matrix C λ ₁ <λ ₂ <λ ₃ , give m ₁ , m ₂ , and m ₃ about λ ₁ , λ ₂ , λ ₃ are expressed as follows:

Define a set of elementary symmetric polynomials s _q about λ ₁ , λ ₂ , λ ₃ as follows:

s ₁ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ +λ ₂ +λ ₃ ,

s ₂ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ λ ₂ +λ ₁ λ ₃ +λ ₂ λ ₃ ,

s ₃ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ λ ₂ λ ₃ . (9)

Substituting equation (9) into equation (8), we get:

s ₁ , s ₂ , s ₃ are obtained by calculating the known m ₁ , m ₂ , m ₃ , and the final estimated eigenvalues λ ₁ , λ ₂ , λ ₃ take s ₁ , s ₂ , s ₃ as coefficients The solution of the polynomial λ ³ -s ₁ λ ² +s ₂ λ - s ₃ =0.

Step 6: Compare the network synchronization domain obtained in step 3 with the eigenvalue spectrum estimated in step 5, and determine the conditions for the network to achieve synchronization stability;

Using the synchronization domain SR obtained in step 3 and the estimated eigenvalues λ ₁ , λ ₂ , and λ ₃ obtained in step 5, let cλ ₁ , cλ ₂ , and cλ ₃ all fall within the scope of the synchronization domain SR, ensuring that the NW small world network stable in the synchronous state.

The beneficial effects produced by the above technical solutions are:

The invention proposes a spectral moment-based NW small-world network system synchronization analysis method, which is different from the existing research on the complex network pinning synchronization problem. Based on random matrix theory, the pinning and synchronization of complex networks is studied from the perspective of NW small-world network dynamic evolution. Stability problem, the conclusions obtained can better reveal the internal mechanism of the interaction between the structural evolution of the actual network system and the dynamic behavior, and then design the actual network system that meets the desired conditions (such as stability and robustness), and predict the network system. It provides theoretical and methodological guidance for behavior and improvement of various performances of real network systems by means of control, and has important economic and social significance.

Description of drawings

Fig. 1 is the flow chart of the NW small world network system synchronization analysis method based on spectral moment of the present invention;

FIG. 2 is a comparison diagram of the eigenvalue distribution of the matrix C and the triangular distribution function fitting the eigenvalue distribution according to the embodiment of the present invention.

Detailed ways

The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

A spectral moment-based synchronization analysis method for NW small-world network systems, as shown in Figure 1, includes the following steps:

Step 1: Construct a NW small world network with N nodes according to the NW small world network model construction algorithm;

是一个对称阵，邻接矩阵A可表示如下：In the NW small world network, each node represents a coupled oscillator, and a network is formed by connecting edges between nodes, and each connecting edge represents the interaction relationship between the coupled oscillators. The network has a symmetrical connection structure and can be used Undirected (unweighted) graph description, its adjacency matrix

is a symmetric matrix, and the adjacency matrix A can be expressed as follows:

表示N×N维实数矩阵的集合，a_ij表示邻接矩阵元素，如果节点i与节点j之间有边相连，则a_ij＝a_ji＝1(i≠j)；否则a_ij＝a_ji＝0(i≠j)；d_i表示节点i的度，记为

D表示度矩阵，记为D＝diag(d_i)；

表示Laplacian(拉普拉斯)矩阵，l_ij表示Laplacian矩阵元素，记为L＝D-A；where N is the number of network nodes,

Represents a set of N×N-dimensional real number matrices, a _ij represents an adjacency matrix element, if there is an edge between node i and node j, then a _ij =a _ji =1 (i≠j); otherwise a _ij =a _ji = 0 (i≠j); d _i represents the degree of node i, denoted as

D represents the degree matrix, denoted as D=diag(d _i );

Represents the Laplacian matrix, and l _ij represents the elements of the Laplacian matrix, denoted as L=DA;

Step 2: Determine the dynamic equation expression of the nodes in the NW small world network system;

For a continuous-time dynamic network composed of N identical nodes, the dynamic equation of the ith node is expressed as:

表示第i个节点的状态变量，对时间t的一阶导数为

f(·)表示单个节点自身动力学函数；

为内耦合矩阵，表示各个节点状态变量之间的耦合关系；常数c>0表示全局耦合强度；in,

Represents the state variable of the ith node, and the first derivative with respect to time t is

f( ) represents the dynamic function of a single node itself;

is the internal coupling matrix, which represents the coupling relationship between the state variables of each node; the constant c>0 represents the global coupling strength;

Step 3: Analyze the synchronization stability of the NW small-world network with the main stability function method, and determine the synchronization and stability region of the NW small-world network system;

The synchronization stability of the NW small-world network, when x ₁ (t)=x ₂ (t)=...=x _N (t)=s(t), means that the NW small-world network achieves synchronization stability, where s(t) is called synchronous state; when pinning control is introduced, the dynamic equation of the controlled NW small-world network is expressed as:

in:

u _i (t)=-cf _i Γ(x _i (t)-s(t)),i=1,2,...,N (3)

同步态s(t)是单个节点系统

的一个解，满足

f _i represents the pinning control gain of the controlled node i, δ _i represents whether the i-th node is controlled or not, if pinning control is applied to node i, then δ _i =1, f _i >0; otherwise δ _i =fi ₌ 0 ; F=diag(f _i ) represents the pinning control gain matrix; l represents the number of pinning control nodes, denoted as

Synchronous state s(t) is a single node system

a solution that satisfies

I_n表示n×n维单位矩阵，

得：When the NW small-world network does not achieve synchronization and stability, there is an error between the state of each node and the synchronization state; the error vector is defined as ε _i (t), then x _i (t)=s(t)+ε _i (t); (2) For linearization and variable substitution, let

In represents an n× _n -dimensional identity matrix,

have to:

Among them, J _f (t) is the Jacobian matrix of f(x(t)) in synchronous state s(t); λ _i is the eigenvalue of matrix C=L+F; P is the equivalent transformation matrix of matrix C , the equivalent form of matrix C is denoted as Q=P ^-1 CP;

According to the main stable function method, the region where the maximum Lyapunov exponent of equation (4) is negative is the synchronous stable region of the NW small-world network system, denoted as SR={cλ _i |L _max (cλ _i )<0};

Step 4: Based on the spectral moment analysis method, according to the probability distribution of the network node degree, calculate the expected value of the first three-order spectral moment of the network matrix C after the introduction of the pinning control, and establish the relationship between the first three-order expected spectral moment and the network structure parameters, The network structure parameters include the number of network nodes N, the edge connection probability p, the pinning control gain f, and the number 1 of pinning control nodes;

The network matrix C is a real symmetric matrix, {λ _i , i=1,2,...,N} is denoted as the eigenvalue set of the matrix C, then the q-order spectral moment of the matrix C is expressed as:

When q=1,

When q≤3, the matrix trace satisfies the commutative law, that is, tr(LLF)=tr(LFL)=tr(FLL), we get:

When q=2,

When q=3,

Since the matrix L=DA, expand the (DA) ^q binomial, and we can get:

where T is the total number of triangles contained in the network.

The NW small-world network model considered in this embodiment starts with a regular graph, and initially is a nearest-neighbor coupled network containing N nodes, where each node is connected to its left and right adjacent k nodes; then randomization is performed. Add connecting edges to avoid self-loops and multiple edges, add an edge to a pair of randomly selected nodes with probability p=r/N, and there is at most one edge between any two different nodes, and each node cannot have an edge connected to itself. Since random addition of edges has little effect on the number of triangles in the network, so

The node degree distribution of the NW small-world network obeys the Poisson distribution, that is,

Among them, d _i represents the degree of node i in the NW small world network, d represents the node degree value, and the parameter r=pN;

Calculate the expected value of the first three origin moments of the node degree, we can get:

Assuming that the first l nodes in the NW small-world network are pinned and controlled, and all control gains are constant f, the expression of the expected value of the first third-order spectral moment of the matrix C is:

For better explanation, in this embodiment, it is assumed that the number of nodes N in the NW small world network is 512, the number of adjacent nodes k connected to each node in the network is 3, and the probability of randomizing the network edge p=r/N is 0.078, that is, the parameter r=4. At the same time, the first l nodes in the network are selected as the pinning control object, where l=5 is assumed, and their node degrees are 8, 9, 9, 12, 7 respectively; and the pinning control strength f is the same, here it is assumed that f=5 . The results of the first third-order spectral moment values of the NW small-world network matrix C calculated by formula (7) and the first third-order spectral moment values of the NW small-world network matrix C obtained by the actual simulation are as follows:

Step 5: use the expected value of the spectral moment obtained in step 4 to construct the eigenvalue spectrum of the piecewise linear function fitting matrix C;

According to the expected value of the first three-order spectral moments obtained in step 4, a triangular distribution function is used to construct the distribution of the piecewise function linear fitting characteristic spectrum;

The expected values of the first three-order spectral moments of the matrix C obtained in step 4 are known, denoted as m ₁ , m ₂ , m ₃ ; in order to estimate the three eigenvalues of the matrix C λ ₁ <λ ₂ <λ ₃ , give m ₁ , The expressions of m ₂ , m ₃ about λ ₁ , λ ₂ , λ ₃ are as follows:

Define a set of elementary symmetric polynomials s _q about λ ₁ , λ ₂ , λ ₃ as follows:

s ₁ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ +λ ₂ +λ ₃ ,

s ₂ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ λ ₂ +λ ₁ λ ₃ +λ ₂ λ ₃ ,

s ₃ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ λ ₂ λ ₃ . (9)

Substituting equation (9) into equation (8), we get:

s ₁ , s ₂ , s ₃ can be calculated through the known m ₁ , m ₂ , m ₃ , and the final estimated eigenvalues λ ₁ , λ ₂ , λ ₃ take s ₁ , s ₂ , s ₃ as coefficients The solution of the polynomial λ ³ -s ₁ λ ² +s ₂ λ - s ₃ =0. The estimated values of the eigenvalues λ ₁ , λ ₂ , and λ ₃ calculated by formula (10) are 0.1306, 11.5915, and 18.4244, respectively. The triangular distribution function constructed by the estimated eigenvalues and the eigenvalue distribution of the matrix C are shown in Figure 2.

Step 6: Compare the network synchronization domain obtained in step 3 with the eigenvalue spectrum estimated in step 5, and determine the conditions for the network to achieve synchronization stability;

Using the synchronization domain SR obtained in step 3 and the estimated eigenvalues λ ₁ , λ ₂ , and λ ₃ obtained in step 5, let cλ ₁ , cλ ₂ , and cλ ₃ all fall within the range of the synchronization domain SR, which can ensure the NW small world The network is stable in the synchronous state.

Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, but not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: The technical solutions described in the foregoing embodiments can still be modified, or some or all of the technical features thereof can be equivalently replaced; and these modifications or replacements do not make the essence of the corresponding technical solutions depart from the scope defined by the claims of the present invention .

Claims (5)

1. a NW small world network system synchronization analysis method based on spectral moment, is characterized in that: comprise the following steps: Step 1: Construct a NW small world network with N nodes according to the NW small world network model construction algorithm; In the NW small world network, each node represents a coupled oscillator, and a network is formed by connecting edges between nodes, and each connecting edge represents the interaction relationship between the coupled oscillators. The network has a symmetrical connection structure. adjacency matrix

is a symmetric matrix, N represents the number of network nodes,

Represents a set of N×N-dimensional real number matrices, a _ij represents an adjacency matrix element, if there is an edge between node i and node j, then a _ij =a _ji =1 (i≠j); otherwise a _ij =a _ji = 0 (i≠j); d _i represents the degree of node i, denoted as

D represents the degree matrix, denoted as D=diag(d _i );

Represents the Laplacian matrix, and l _ij represents the elements of the Laplacian matrix, denoted as L=DA; Step 2: Determine the dynamic equation expression of the nodes in the NW small world network system; For a continuous-time dynamic network composed of N identical nodes, the dynamic equation of the ith node is expressed as:

in,

Represents the state variable of the ith node, and the first derivative with respect to time t is

f( ) represents the dynamic function of a single node itself;

is the internal coupling matrix, which represents the coupling relationship between the state variables of each node; the constant c>0 represents the global coupling strength; Step 3: Analyze the synchronization stability of the NW small-world network with the main stability function method, and determine the synchronization and stability region of the NW small-world network system; Step 4: Based on the spectral moment analysis method, according to the probability distribution of the network node degree, calculate the expected value of the first three-order spectral moment of the network matrix C after the introduction of the pinning control, and establish the relationship between the first three-order expected spectral moment and the network structure parameters, The network structure parameters include the number of network nodes N, the edge connection probability p, the pinning control gain f, and the number 1 of pinning control nodes; Step 5: use the expected value of the spectral moment obtained in step 4 to construct the eigenvalue spectrum of the piecewise linear function fitting matrix C; According to the expected value of the first three-order spectral moments obtained in step 4, a triangular distribution function is used to construct the distribution of the piecewise function linear fitting characteristic spectrum; Step 6: Compare the network synchronization domain obtained in step 3 with the eigenvalue spectrum estimated in step 5, and determine the conditions for the network to achieve synchronization stability. 2. a kind of NW small world network system synchronization analysis method based on spectral moment according to claim 1, is characterized in that, the synchronization stability of NW small world network described in step 3, when x ₁ (t)=x ₂ (t)=...=x _N (t)=s(t), it means that the NW small-world network achieves synchronization and stability, where s(t) is called the synchronization state; if pinning control is introduced, the controlled NW is small. The dynamic equation of the world network is expressed as:

in: u _i (t)=-cf _i Γ(x _i (t)-s(t)),i=1,2,...,N (3) f _i represents the pinning control gain of the controlled node i, δ _i represents whether the i-th node is controlled or not, if pinning control is applied to node i, then δ _i =1, f _i >0; otherwise δ _i =fi ₌ 0 ; F=diag(f _i ) represents the pinning control gain matrix; l represents the number of pinning control nodes, denoted as

Synchronous state s(t) is a single node system

a solution that satisfies

When the NW small-world network does not achieve synchronization and stability, there is an error between the state of each node and the synchronization state; the error vector is defined as ε _i (t), then x _i (t)=s(t)+ε _i (t); (2) For linearization and variable substitution, let

In represents an n× _n -dimensional identity matrix,

have to:

Among them, J _f (t) is the Jacobian matrix of f(x(t)) in synchronous state s(t); λ _i is the eigenvalue of matrix C=L+F; P is the equivalent transformation matrix of matrix C , the equivalent form of matrix C is denoted as Q=P ^-1 CP; According to the main stable function method, the region where the maximum Lyapunov exponent of equation (4) is negative is the synchronous stable region of the NW small-world network system, denoted as SR={cλ _i |L _max (cλ _i )<0}. 3. a kind of NW small-world network system synchronization analysis method based on spectral moment according to claim 1, is characterized in that, described in step 4, network matrix C is a real symmetric matrix, {λ _i , i=1, 2,...,N} is denoted as the set of eigenvalues of matrix C, then the q-order spectral moment of matrix C is expressed as:

At the initial moment, the network is a nearest-neighbor coupled network composed of N nodes, and the degree of each node is 2k; on the basis of this network, random edges are connected with probability p (to avoid self-loops and multiple edges), and an NW small world network is obtained. , and its node degree distribution obeys a Poisson distribution, that is,

Among them, d _i represents the degree of node i in the NW small world network, d represents the node degree value, and the parameter r=pN; Assuming that the first l nodes in the NW small-world network are pinned and controlled, and all control gains are constant f, the expression of the expected value of the first third-order spectral moment of the matrix C is:

4. a kind of spectral moment-based NW small-world network system synchronization analysis method according to claim 1, is characterized in that, described in step 5, the first third-order spectral moment expectation value is denoted as m ₁ , m ₂ , m ₃ ; In order to estimate the three eigenvalues λ ₁ <λ ₂ <λ ₃ of the matrix C, the expressions of m ₁ , m ₂ , m ₃ about λ ₁ , λ ₂ , λ ₃ are given as follows:

Define a set of elementary symmetric polynomials s _q about λ ₁ , λ ₂ , λ ₃ as follows: s ₁ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ +λ ₂ +λ ₃ , s ₂ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ λ ₂ +λ ₁ λ ₃ +λ ₂ λ ₃ , s ₃ (λ ₁ ,λ ₂ ,λ ₃ )=λ ₁ λ ₂ λ ₃ . (9) Substituting equation (9) into equation (8), we get: s ₁ =3m ₁ ,

s ₁ , s ₂ , s ₃ are obtained by calculating the known m ₁ , m ₂ , m ₃ , and the final estimated eigenvalues λ ₁ , λ ₂ , λ ₃ take s ₁ , s ₂ , s ₃ as coefficients The solution of the polynomial λ ³ -s ₁ λ ² +s ₂ λ - s ₃ =0. 5. a kind of spectral moment-based NW small world network system synchronization analysis method according to claim 1, is characterized in that, described in step 6, compares the network synchronization domain that step 3 obtains and the eigenvalue estimated in step 5 spectrum, using the synchronization domain SR obtained in step 3 and the estimated eigenvalues λ ₁ , λ ₂ , and λ ₃ obtained in step 5, so that cλ ₁ , cλ ₂ , and cλ ₃ all fall within the synchronization domain SR range to ensure NW The small world network is stable in the synchronous state.

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