CN111884849B - Random network system containment synchronization stability analysis method based on spectrum moment - Google Patents

Random network system containment synchronization stability analysis method based on spectrum moment Download PDF

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CN111884849B
CN111884849B CN202010717851.3A CN202010717851A CN111884849B CN 111884849 B CN111884849 B CN 111884849B CN 202010717851 A CN202010717851 A CN 202010717851A CN 111884849 B CN111884849 B CN 111884849B
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CN111884849A (en
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项林英
余言英
邬思宏
韩冰心
陈飞
郭戈
武艳芝
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Northeastern University Qinhuangdao Branch
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Abstract

The invention provides a random network system containment synchronization stability analysis method based on spectral moments, and relates to the technical field of control and information. The invention is based on the random matrix theory, researches the problem of the containment synchronization stability of the complex network from the perspective of the dynamic evolution of the actual network structure, establishes the qualitative and quantitative relations between the containment synchronization of the random network system and the node dynamics, the network structure and the control strategy by analyzing the expected spectral moment and the network structure parameters of the matrix C of the random network controlled by containment, further enriches and perfects the control theory of the complex network, is helpful for us to further understand the evolution law and the dynamics mechanism of the complex network system structure in the real world, further provides theoretical and methodical guidance for designing the actual network meeting the expected conditions and improving various performances of the actual network system, and has important economic and social meanings.

Description

Random network system containment synchronization stability analysis method based on spectrum moment
Technical Field
The invention relates to the technical field of control and information, in particular to a random network system containment synchronization stability analysis method based on spectrum moment.
Background
In the last two decades, the synchronization problem of complex dynamic networks has been receiving more and more attention, wherein an important problem is to determine the synchronization capability of the network, i.e. the range of global coupling strength required for achieving network synchronization. The method is characterized in that partial nodes in the network are controlled, so that all the nodes of the complex network tend to be in the same state, and the problem of the containment synchronization of the complex network is solved. In the existing research on the problem of the complicated network containment synchronization, a static network with a fixed structure obtained after a certain time evolution is mostly assumed to be a network, the dynamic evolution characteristic of the network structure and the randomness of the connection edges are not directly reflected in a dynamic system model, and the corresponding research result is greatly limited in practical application, so that a new analysis method is needed to fully reflect the dynamic evolution characteristic of the actual network and the randomness generated by the connection edges.
Based on the random matrix theory, the problem of the containment synchronization stability of the complex network is researched from the perspective of the dynamic evolution of the actual network structure, which is helpful for further understanding the evolution rule and the dynamics mechanism of the complex network system structure in the real world, and further provides theoretical and methodical guidance for designing the actual network meeting the expected conditions and improving various performances of the actual network system, and has important economic and social meanings.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a random network system containment synchronization stability analysis method based on spectral moments.
The technical scheme adopted by the invention is as follows:
a random network system containment synchronization stability analysis method based on spectrum moment comprises the following steps:
step 1: constructing a random network system with N nodes according to a random graph model construction algorithm;
in the random network system, each node represents a coupled oscillator, a network is formed by nodes through connecting edges, each connecting edge represents the interaction relation between the connected coupled oscillators, and an adjacent matrix A is (A)ij)∈RN×NIs a symmetric array, AijRepresenting the contiguous matrix elements, RN×NRepresenting a set of NxN dimensional real number matrixes, wherein N represents the number of random network nodes; in the adjacency matrix A, if edges are connected between the nodes i and j, Aij＝Aji1(j ≠ i); otherwise Aij＝Aji0(j ≠ i); wherein d isiDegree representing node i, denoted
Figure BDA0002598858320000011
D denotes the degree matrix, denoted D ═ diag (D)i)；L＝(Lij)∈RN×NRepresents a Laplacian matrix, LijRepresents Laplacian matrix element, and is marked as L-D-A;
step 2: determining a dynamic equation expression of a node in a topological graph of a random network system;
the continuous time dynamic network is composed of N same nodes, and the kinetic equation of the ith node is expressed as follows:
Figure BDA0002598858320000021
wherein x isi(t)＝[xi1(t),xi2(t),...,xin(t)]T∈RnThe state variable representing the ith node, the first derivative with respect to time t being
Figure BDA0002598858320000022
f (-) represents a single node self dynamic function; h is belonged to Rn×nThe node state variables are coupled in an inner coupling matrix; the constant α > 0 represents the global coupling strength; l ═ L (L)ij)∈RN×NIs a Laplacian matrix.
And step 3: analyzing the synchronous stability of the random network by using a main stability function method, and determining a synchronous stability area of a random network system;
step 3.1: calculating a kinetic equation of the random network under the control of the constraint:
Figure BDA0002598858320000023
among them are: u. ofi(t)＝-αbiH(xi(t)-s(t))，i＝1,2,...,N (3)
biRepresenting the holddown control gain, δ, of the controlled node iiIndicating whether the ith node is controlled or not, if the pinning control is applied to the node i, deltai＝1，bi>0; else δi＝bi0; l represents the number of holdback control nodes, and is recorded as
Figure BDA0002598858320000024
Target state s (t) single node system
Figure BDA0002598858320000025
A solution of (a) satisfies
Figure BDA0002598858320000026
B＝diag(bi) A holddown control gain matrix is represented.
The control of the containment is that only l (l is more than or equal to 1 and less than N) nodes in the network are directly controlled, so that the whole network is stable in a synchronous state;
step 3.2: defining an error vector as εi(t)＝xi(t) -s (t), linearizing the formula (2) and performing variable substitution, and allowing
Figure BDA0002598858320000027
InRepresenting an n x n dimensional identity matrix,
Figure BDA0002598858320000028
obtaining:
Figure BDA0002598858320000029
wherein, Jf(s (t)) is a Jacobian matrix of f (x (t)) in a target state s (t), λiIs the eigenvalue of matrix C, which is L + B; p is an approximate conversion matrix of the matrix C, and the approximate type of the matrix C is marked as Q ═ P-1CP；
From the main stabilization function method, the maximum Lyapunov (Lyapunov) exponent Λ (α λ) of the formula (4)i) Determining the zero solution stability of the equation and the maximum Lyapunov exponent Λ (α λ)i) The negative region is the sync region and is denoted as SR ═ α λi|Λ(αλi) < 0 }; for a synchronous domain bounded dipole system, there is 0 < alpha1＜α2，α1,α2Respectively representing the minimum and maximum coupling strengths for synchronously stabilizing the vibrator system; when alpha is1＜αλi＜α2Time, main stabilization function Λ (α λ)i) If the value is less than 0, the synchronous stable area of the random network system is obtained;
and 4, step 4: obtaining an expected spectral moment of a random network matrix C under the control of the containment based on spectral moment analysis, and establishing the first three-order expected spectral moment and network structure parameters including the relation among the number N of random network nodes, the probability p of connecting edges, containment control gains and the number l of containment control nodes;
the random network matrix C is a real symmetric matrix, { lambda }i1, 2.. N } is recorded as the eigenvalue set of the matrix C, then the k-order spectral moments of the matrix C are represented as:
Figure BDA0002598858320000031
let p be the probability of connecting edges between any two different nodes of the random network, and the random network is marked as G (N, p); the probability distribution of the degree of the random network nodes follows a binomial distribution, i.e.
Figure BDA0002598858320000032
Wherein d isiRepresenting the degree of a node i in the random network, and d represents a node value; when the number of random network nodes tends to infinity and the product of N and p is constant, the node degree follows Poisson distribution, i.e. the node degree
Figure BDA0002598858320000033
When N → ∞ assumes that the i nodes in the network are controlled and that all control gains are constant b, the expression for the first three orders of the desired spectral moments of matrix C is:
Figure BDA0002598858320000034
when l > 1, selecting the neighbor node of the node with the highest degree in the network to exert the control of the constraint and l is a known quantity;
and 5: calculating expected values of the first three-order expected spectral moments, and fitting a characteristic value spectrum of the random network matrix C by using a piecewise linear function;
fitting a characteristic spectrum of the matrix C by using a Piecewise Linear-Linear Reconstruction (Piecewise-Linear Reconstruction), and using a triangular Reconstruction method, knowing that the first three-order expected spectral moment of the matrix C is marked as { m }1,m2,m33 eigenvalues of the matrix are denoted as { λ }1,λ2,λ3Are and λ1＜λ2＜λ3Then { m1,m2,m3With respect to lambda1＜λ2＜λ3Is expressed as
Figure BDA0002598858320000041
Defining information about a characteristic value { lambda1,λ2,λ3The elementary symmetric polynomial of { y }1,y2,y3Is expressed as
Figure BDA0002598858320000042
Bringing it into formula (9) to obtain
Figure BDA0002598858320000043
According to known quantity m1,m2,m3Calculate { y }1,y2,y3The value of the characteristic value { lambda } is finally obtained1,λ2,λ3Is { y } is1,y2,y3Polynomial lambda of coefficient3-y1λ2+y2λ-y30 root.
Step 6: comparing the synchronous stable region in the step 3 with the characteristic value spectrum in the step 5, and determining the range of the overall coupling strength for stabilizing and synchronizing the network;
judging the stability of the network synchronization state, and verifying whether the synchronization area of the network contains (alpha lambda)1,αλ3) If yes, then the method is true. Thus, for the coupled oscillator network, there is a synchronization region (α)1,α2) When global coupling strength of the network
Figure BDA0002598858320000044
Then the network achieves synchronization.
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in:
the invention provides a random network system containment synchronization stability analysis method based on spectral moments, which is different from the traditional research on the containment synchronization problem of a complex network in the greatest way.
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FIG. 1 is a flow chart of a method for analyzing the holddown synchronization stability of a stochastic network system based on spectral moments according to the present invention;
FIG. 2 is a histogram of eigenvalues of a matrix C and trigonometric functions consistent with the desired spectral moments, in accordance with an embodiment of the present invention.
Detailed Description
The following detailed description of embodiments of the invention refers to the accompanying drawings.
In this embodiment, a flow chart of a method for analyzing the holddown synchronization stability of a random network system based on spectrum moments, as shown in fig. 1, includes the following steps:
step 1: constructing a random network with N nodes according to a random graph model construction algorithm, wherein each node in the network represents a coupled oscillator, and each edge represents an interaction relation between the connected coupled oscillators;
in the random network system, nodes form a network through connecting edges, here, an unweighted undirected network is considered, and an adjacent matrix A is (A)ij)∈RN×NIs a symmetric array, AijRepresenting the contiguous matrix elements, RN×NRepresenting a set of NxN dimensional real number matrixes, wherein N represents the number of random network nodes; the adjacency matrix a is defined as follows: if there is an edge connection between node i and node j, then Aij＝Aji1(j ≠ i); otherwise Aij＝Aji＝0(j≠i)；diDegree representing node i, denoted
Figure BDA0002598858320000051
D denotes the degree matrix, denoted D ═ diag (D)i)；L＝(Lij)∈RN×NRepresents a Laplacian matrix, LijRepresents Laplacian matrix element, and is marked as L-D-A;
step 2: determining a dynamic equation expression of a node in a random network topological graph;
considering a continuous-time dynamic network composed of N identical nodes, the kinetic equation of the ith node is expressed as:
Figure BDA0002598858320000052
wherein x isi(t)＝[xi1(t),xi2(t),...,xin(t)]T∈RnThe state variable representing the ith node, the first derivative with respect to time t being
Figure BDA0002598858320000057
f (-) represents a single node self dynamic function; h is belonged to Rn×nThe node state variables are coupled in an inner coupling matrix; the constant α > 0 represents the global coupling strength; l ═ L (L)ij)∈RN×NIs Lan aplian matrix.
And step 3: analyzing the synchronous stability of the random network by using a main stability function method, and determining a synchronous stability area of a random network system;
step 3.1: the kinetic equation of the random network controlled by the constraint is expressed as
Figure BDA0002598858320000053
Among them are: u. ofi(t)＝-αbiH(xi(t)-s(t))，i＝1,2,...,N (3)
biRepresents the tie-down control gain of the controlled node i, and if tie-down control is applied to the node i, deltai＝1，bi>0; else δi＝bi0; l represents the number of holdback control nodes, and is recorded as
Figure BDA0002598858320000054
Target state s (t) single node system
Figure BDA0002598858320000055
A solution of (a) satisfies
Figure BDA0002598858320000056
B＝diag(bi) A holddown control gain matrix is represented.
Step 3.2: defining an error vector as εi(t)＝xi(t) -s (t), linearizing the formula (2) and performing variable substitution, and allowing
Figure BDA0002598858320000061
InRepresenting an n x n dimensional identity matrix,
Figure BDA0002598858320000062
obtaining:
Figure BDA0002598858320000063
wherein, Jf(s (t)) is a Jacobian matrix of f (x (t)) in a target state s (t), λiIs the eigenvalue of matrix C, which is L + B; p is an approximate conversion matrix of the matrix C, and the approximate type of the matrix C is marked as Q ═ P-1CP；
The zero solution stability of equation (4) is determined by the maximum Lyapunov exponent Λ (α λ) of the equationi) Determining; according to the Master Stability Function method (Master Stability Function), the region with the negative maximum Lyapunov exponent in equation (4) is the synchronization region, and is marked as SR ═ α λi|Λ(αλi) < 0 }; considering a network system with nodes as oscillators, for a oscillator system with a bounded synchronization domain, 0 < alpha exists1＜α2，α1,α2Respectively representing the minimum and maximum coupling strengths for synchronously stabilizing the vibrator system; when alpha is1＜αλi＜α2Time, main stabilization function Λ (α λ)i)＜0；
And 4, step 4: obtaining an expected spectral moment of a random network matrix C under the control of the containment based on spectral moment analysis, and establishing the first three-order expected spectral moment and network structure parameters, including the number N of nodes of a random network, the probability p of a connecting edge between any two different nodes of the random network, the containment control gain and the number l of the containment control nodes;
the control is directly applied to l (1 is more than or equal to l and less than N) nodes in the network, so that the whole network is stable in a synchronous state.
The matrix C is a real symmetric matrix, { lambda }, of the weightless and directionless networki1, 2.. N } is recorded as the eigenvalue set of the matrix C, then the k-order spectral moments of the matrix C are represented as:
Figure BDA0002598858320000064
when k is equal to 1, the first step is carried out,
Figure BDA0002598858320000065
when k ≦ 3, the matrix trace operation satisfies the commutative law, e.g., tr (llb) tr (lbl) tr (B)LL), and by a binomial expansion of the matrix
Figure BDA0002598858320000066
Obtaining:
Figure BDA0002598858320000067
and is composed of the identity
Figure BDA0002598858320000068
Obtaining:
Figure BDA0002598858320000069
when k is equal to 2, the number of the bits is increased,
Figure BDA00025988583200000610
when k is 3, the number of the groups is 3,
Figure BDA0002598858320000071
from the existing documents V.M.Preciado and A.Jadbabaie.moment-based analysis of synchronization in small-world networks of interactions.48th IEEE Conference on Decision and Control and 28th Chinese Control Conference,2009, pp.1690-1695, Lemma 2 is knownii＝0，(A2)ii＝di，(A3)ii＝2tiWherein, tiThe number of triangles associated with node i; since the matrix L is D-A, will (D-A)kThe second term is developed and finished to obtain
Lii＝(D-A)ii＝di，
Figure BDA0002598858320000072
Figure BDA0002598858320000073
The matrix B is diagonal, so
Figure BDA0002598858320000074
To sum up to get
Figure BDA0002598858320000075
Step 4.2: let p be the probability of connecting edges between any two different nodes of the random network, and the random network is marked as G (N, p); the probability distribution of the degree of the random network nodes follows a binomial distribution, i.e.
Figure BDA0002598858320000076
Wherein d isiRepresenting the degree of a node i in the random network, and d represents a node value; when the number of random network nodes tends to infinity and the product of N and p is constant, the node degree follows Poisson distribution, i.e. the node degree
Figure BDA0002598858320000077
Calculating an expected origin moment of the random network node degree to obtain:
Figure BDA0002598858320000078
Figure BDA0002598858320000079
Figure BDA00025988583200000710
let TNFor the number of all triangles in the random network, i.e.
Figure BDA00025988583200000711
When p is more than 0 and less than or equal to 1, memory i, j connected indicative functionIs composed of
Figure BDA00025988583200000712
Wherein e isijRepresenting an edge between nodes I and j, E is an edge set, and IijAre independent of each other. We give TNIs expected to
Figure BDA0002598858320000081
Is calculated to obtain
Figure BDA0002598858320000082
When N → ∞ is assumed to control randomly selected l nodes in the network, and all control gains are constants b greater than 0, the expression of the first three orders of the desired spectral moments of matrix C is:
Figure BDA0002598858320000083
when l > 1, the neighbor node of the node in the network with the highest degree is selected to exert the containment control and l is a known quantity.
And 5: calculating expected values of the first three-order expected spectral moments, and fitting a characteristic value spectrum of the random network matrix C by using a piecewise linear function;
fitting a characteristic spectrum of the matrix C by using a Piecewise Linear-Linear Reconstruction (Piecewise-Linear Reconstruction), and using a triangular Reconstruction method, knowing that the first three-order expected spectral moment of the matrix C is marked as { m }1,m2,m33 eigenvalues of the matrix are denoted as { λ }1,λ2,λ3Are and λ1＜λ2＜λ3Then { m1,m2,m3With respect to lambda1＜λ2＜λ3Is expressed as
Figure BDA0002598858320000084
Defining information about a characteristic value { lambda1,λ2,λ3The elementary symmetric polynomial of { y }1,y2,y3Is expressed as
Figure BDA0002598858320000085
Bringing it into formula (9) to obtain
Figure BDA0002598858320000086
According to known quantity m1,m2,m3Calculate { y }1,y2,y3The value of the characteristic value { lambda } is finally obtained1,λ2,λ3Is { y } is1,y2,y3Polynomial lambda of coefficient3-y1λ2+y2λ-y30 root.
In order to better describe the present invention, in this embodiment, the number N of random network nodes is 512, the probability p of edge connection is 0.02, the number l of control nodes is 10, and the control gain b is 9, so as to perform random control, that is, the first 10 nodes are taken as controlled nodes, and the node degrees thereof are 20, 16, 18, 26, 22, 17, 13, 21, 33, and 25, respectively. Calculate the first third order desired moment { m } of matrix C1,m2,m3Is {20.6,462.7,11050}, and the feature value of the triangle reconstruction is { λ1,λ2,λ3Is {5.4,21.4,35.1 }. FIG. 2 is a histogram of eigenvalues of a trigonometric function and matrix C corresponding to desired spectral moments;
step 6: comparing the synchronous stable region in the step 3 with the characteristic value spectrum in the step 5, and determining the range of the overall coupling strength for stabilizing and synchronizing the network;
judging the stability of the network synchronization state, verifying whether the synchronization area of the network contains(αλ1,αλ3) If yes, then the method is true. Thus, for the coupled oscillator network, there is a synchronization region (α)1,α2) When global coupling strength of the network
Figure BDA0002598858320000091
Then the network achieves synchronization.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; such modifications and substitutions do not depart from the spirit of the corresponding technical solutions and scope of the present invention as defined in the appended claims.

Claims (4)

1. A random network system containment synchronization stability analysis method based on spectrum moment is characterized in that: the method comprises the following steps:
step 1: constructing a random network system with N nodes according to a random graph model construction algorithm;
in the random network system, each node represents a coupled oscillator, a network is formed by nodes through connecting edges, each connecting edge represents the interaction relation between the connected coupled oscillators, and an adjacent matrix A is (A)ij)∈RN×NIs a symmetric array, AijRepresenting the contiguous matrix elements, RN×NRepresenting a set of NxN dimensional real number matrixes, wherein N represents the number of random network nodes; in the adjacency matrix A, if edges are connected between the nodes i and j, Aij＝Aji1, j ≠ i; otherwise Aij＝Aji0, j is not equal to i; wherein d isiDegree representing node i, denoted
Figure FDA0003506821190000011
D denotes the degree matrix, denoted D ═ diag (D)i)；L＝(Lij)∈RN ×NRepresents a Laplacian matrix, LijRepresents Laplacian matrix element, and is marked as L-D-A;
step 2: determining a dynamic equation expression of a node in a topological graph of a random network system;
the continuous time dynamic network is composed of N same nodes, and the kinetic equation of the ith node is expressed as follows:
Figure FDA0003506821190000012
wherein x isi(t)＝[xi1(t),xi2(t),...,xin(t)]T∈RnThe state variable representing the ith node, the first derivative with respect to time t being
Figure FDA0003506821190000013
f (-) represents a single node self dynamic function; h is belonged to Rn×nThe node state variables are coupled in an inner coupling matrix; the constant α > 0 represents the global coupling strength; l ═ L (L)ij)∈RN×NIs a Laplacian matrix;
and step 3: analyzing the synchronous stability of the random network by using a main stability function method, and determining a synchronous stability area of a random network system;
step 3.1: calculating a kinetic equation of the random network under the control of the constraint:
Figure FDA0003506821190000014
among them are: u. ofi(t)＝-αbiH(xi(t)-s(t))，i＝1,2,...,N (3)
biRepresenting the holddown control gain, δ, of the controlled node iiIndicating whether the ith node is controlled or not, if the pinning control is applied to the node i, deltai＝1，bi>0; else δi＝bi0; l represents a holddown control sectionNumber of points, note
Figure FDA0003506821190000015
Target state s (t) single node system
Figure FDA0003506821190000016
A solution of (a) satisfies
Figure FDA0003506821190000017
B＝diag(bi) Representing a holddown control gain matrix;
step 3.2: defining an error vector as εi(t)＝xi(t) -s (t), linearizing the formula (2) and performing variable substitution, and allowing
Figure FDA0003506821190000018
InRepresenting an n x n dimensional identity matrix,
Figure FDA0003506821190000019
obtaining:
Figure FDA0003506821190000021
wherein, Jf(s (t)) is a Jacobian matrix of f (x (t)) in a target state s (t), λiIs the eigenvalue of matrix C, which is L + B; p is an approximate conversion matrix of the matrix C, and the approximate type of the matrix C is marked as Q ═ P-1CP；
According to the method of the main stabilization function, the maximum Lyapunov exponent Λ (alpha lambda) of the formula (4)i) Determining the zero solution stability of the equation and the maximum Lyapunov exponent Λ (α λ)i) The negative region is the sync region and is denoted as SR ═ α λi|Λ(αλi) < 0 }; for a synchronous domain bounded dipole system, there is 0 < alpha1＜α2，α1,α2Respectively representing the minimum and maximum coupling strengths for synchronously stabilizing the vibrator system; when alpha is1＜αλi＜α2Time, main stabilization function Λ (α λ)i) If the value is less than 0, the synchronous stable area of the random network system is obtained;
and 4, step 4: obtaining an expected spectral moment of a random network matrix C under the control of the containment based on spectral moment analysis, and establishing the first three-order expected spectral moment and network structure parameters including the relation among the number N of random network nodes, the probability p of connecting edges, containment control gains and the number l of containment control nodes;
and 5: calculating expected values of the first three-order expected spectral moments, and fitting a characteristic value spectrum of the random network matrix C by using a piecewise linear function;
step 6: comparing the synchronous stable region in the step 3 with the characteristic value spectrum in the step 5, and determining the range of the overall coupling strength for stabilizing and synchronizing the network;
judging the stability of the network synchronization state, and verifying whether the synchronization area of the network contains (alpha lambda)1,αλ3) If yes, then it is true; thus, for the coupled oscillator network, there is a synchronization region (α)1,α2) When global coupling strength of the network
Figure FDA0003506821190000022
Then the network achieves synchronization.
2. The method for analyzing the holdover synchronization stability of the random network system based on the spectral moments as claimed in claim 1, wherein the holdover control in step 3.1 is to directly apply control to only l (1 ≦ l < N) nodes in the network, so that the whole network is stable in the synchronization state.
3. The method for analyzing the holdover synchronization stability of the random network system based on the spectral moments as claimed in claim 1, wherein the random network matrix C in the step 4 is a real symmetric matrix, { λ }i1, 2.. N } is recorded as the eigenvalue set of the matrix C, then the k-order spectral moments of the matrix C are represented as:
Figure FDA0003506821190000023
let p be the probability of connecting edges between any two different nodes of the random network, and the random network is marked as G (N, p); the probability distribution of the degree of the random network nodes follows a binomial distribution, i.e.
Figure FDA0003506821190000031
Wherein d isiRepresenting the degree of a node i in the random network, and d represents a node value; when the number of random network nodes tends to infinity and the product of N and p is constant, the node degree follows Poisson distribution, i.e. the node degree
Figure FDA0003506821190000032
When N → ∞ assumes that the i nodes in the network are controlled and that all control gains are constant b, the expression for the first three orders of the desired spectral moments of matrix C is:
Figure FDA0003506821190000033
when l > 1, the neighbor node of the node in the network with the highest degree is selected to exert the containment control and l is a known quantity.
4. The method as claimed in claim 1, wherein the eigenvalue spectrum of the fitted random network matrix C in step 5 is a piecewise linear function, and a triangular reconstruction method is used, so that the first three expected spectral moments of the known matrix C are denoted as { m } m1,m2,m33 eigenvalues of the matrix are denoted as { λ }1,λ2,λ3Are and λ1＜λ2＜λ3Then { m1,m2,m3With respect to lambda1＜λ2＜λ3Is expressed as
Figure FDA0003506821190000034
Defining information about a characteristic value { lambda1,λ2,λ3The elementary symmetric polynomial of { y }1,y2,y3Is expressed as
Figure FDA0003506821190000035
Bringing it into formula (9) to obtain
Figure FDA0003506821190000036
According to known quantity m1,m2,m3Calculate { y }1,y2,y3The value of the characteristic value { lambda } is finally obtained1,λ2,λ3Is { y } is1,y2,y3Polynomial lambda of coefficient3-y1λ2+y2λ-y30 root.
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