CN112464414A - NW small-world network system synchronous analysis method based on spectrum moment - Google Patents
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Abstract
The invention provides a synchronous analysis method of a NW (NW) small-world network system based on a spectrum moment, and relates to the technical field of control and information. The method is based on a random matrix theory, researches the problem of the containment synchronization stability of the complex network from the view of the dynamic evolution of the NW small-world network, and the obtained conclusion can reveal the internal mechanism principle that the structure evolution and the dynamic behavior of the actual network system influence each other, thereby providing theoretical and methodical guidance for designing the actual network system meeting expected conditions (such as stability and robustness), predicting the behavior of the network system and improving various performances of the actual network system by a control means, determining the conditions for leading the network to achieve the synchronization stability, and having important economic and social meanings.
Description
Technical Field
The invention relates to the technical field of control and information, in particular to a synchronous analysis method of a NW (NW) small-world network system based on a spectrum moment.
Background
The synchronization phenomenon of complex networks is widely existed in the nature and human society, and has been receiving more and more attention in recent years. The control is applied to part of nodes in the network, so that all the nodes in the network tend to be in the same state, which is called the problem of containment synchronization of a complex network. In the existing research on the problem of the complicated network containment synchronization, a network is mostly assumed to be a deterministic network, and the randomness of the topological structure evolution in an actual network system is not fully considered. Neglecting the network synchronization analysis research result of generating randomness by connecting edges is greatly limited in practical application. Therefore, a new network synchronization analysis method capable of fully reflecting the randomness of the evolution of the actual network topology structure and the randomness of the edge generation is needed to be developed.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a synchronous analysis method of an NW small-world network system based on spectral moments. The invention is based on the random matrix theory, considers the characteristic that the actual network system has randomized connection edges, is beneficial to understanding the evolution law of the complex network system structure in the real world and the dynamics mechanism thereof, and further provides theoretical and methodical guidance for designing the actual network system meeting expected conditions, predicting the behavior of the network system and improving various performances of the actual network system by control means.
The technical scheme adopted by the invention is as follows:
a synchronous analysis method of an NW small-world network system based on spectral moments comprises the following steps:
step 1: constructing an NW small-world network with N nodes according to an NW small-world network model construction algorithm;
in the NW small-world network, each node represents a coupled oscillator, a network is formed by connecting edges among the nodes, and each connecting edge represents the coupled oscillators connected among the coupled oscillatorsThe network has a symmetrical connection structure, which is adjacent to the matrix
Is a symmetric array, N represents the number of network nodes,
representing a set of NxN-dimensional real matrices, aijRepresenting the adjacency matrix element, a if there is an edge connection between node i and node jij＝aji1(i ≠ j); otherwise aij＝aji＝0(i≠j)；diDegree representing node i, denoted
D denotes the degree matrix, denoted D ═ diag (D)i)；
Represents 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 an NW small-world 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:
wherein,
the state variable representing the ith node, the first derivative with respect to time t being
f (-) represents a single node self dynamic function;
the node state variables are coupled in an inner coupling matrix; constant c>0 represents the global coupling strength;
and step 3: analyzing the synchronous stability of the NW small-world network by using a main stability function method, and determining a synchronous stable region of the NW small-world network system;
synchronization stability of the NW small world network, when x1(t)＝x2(t)＝...＝xN(t) s (t), which indicates that the NW small-world network achieves synchronization stability, where s (t) is called a synchronization state; with the introduction of the containment control, the dynamical equations of the controlled NW small-world network are expressed as:
wherein:
ui(t)＝-cfiΓ(xi(t)-s(t)),i＝1,2,...,N (3)
firepresenting 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，fiIs greater than 0; else δi＝fi＝0；F＝diag(fi) Representing a holddown control gain matrix; l represents the number of holdback control nodes, and is recorded as
The synchronization state s (t) being a single node system
A solution of (a) satisfies
When the NW small-world network does not reach the synchronous stability, the state of each node has an error with the synchronous state; defining an error vector as εi(t), then xi(t)＝s(t)+εi(t); linearizing and replacing variables of the formula (2)
InRepresenting an n x n dimensional identity matrix,
obtaining:
wherein, Jf(t) is the Jacobian matrix of f (x (t)) in the sync state s (t); lambda [ alpha ]iIs the eigenvalue of matrix C ═ L + F; 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 main stability function method, the region where the maximum Lyapunov exponent of equation (4) is negative is the synchronous stability region of the NW small-world network system, and is denoted as SR ═ c λi|Lmax(cλi)＜0}；
And 4, step 4: based on a spectral moment analysis method, calculating the expected values of the first three spectral moments of the network matrix C after the containment control is introduced according to the probability distribution of the network node degrees, and establishing the relation between the expected spectral moments of the first three and network structure parameters, wherein the network structure parameters comprise the number N of network nodes, the probability p of connecting edges, the containment control gain f and the number l of containment control nodes;
the network matrix C is a real symmetric matrix, { lambda }i1, 2.. N } is recorded as the eigenvalue set of matrix C, then the q-order spectral moments of matrix C are represented as:
the initial time network is a nearest neighbor coupling network consisting of N nodes, and the degree of each node is 2 k; random edge connection (avoiding self-loop and multiple edges) is carried out on the basis of the network by using probability p, so that an NW small-world network is obtained, and the node degree distribution of the NW small-world network is subjected to Poisson distribution, namely
Wherein d isiRepresenting the degree of a node i in the NW small-world network, d represents a node value, and a parameter r is equal to pN;
assuming that the first l nodes in the NW small-world network are subjected to containment control, and all control gains are constants f, the expression of the expected value of the first third moment of spectrum of the matrix C at this time is:
and 5: constructing a characteristic value spectrum of the piecewise linear function fitting matrix C by using the expected value of the spectrum moment obtained in the step 4;
building the distribution of a piecewise function linear fitting characteristic spectrum by adopting a triangular distribution function according to the expected value of the first third moment of spectrum obtained in the step 4;
the expected value of the first third moment is recorded as m1,m2,m3(ii) a To estimate the 3 eigenvalues λ of the matrix C1＜λ2＜λ3Giving m1,m2,m3About lambda1,λ2,λ3The expression of (a) is as follows:
definition of the relation lambda1,λ2,λ3A set of elementary symmetric polynomials sqThe following were used:
s1(λ1,λ2,λ3)＝λ1+λ2+λ3,
s2(λ1,λ2,λ3)＝λ1λ2+λ1λ3+λ2λ3,
s3(λ1,λ2,λ3)＝λ1λ2λ3. (9)
substituting formula (9) for formula (8) to obtain:
by known m1,m2,m3Is calculated to obtain s1,s2,s3Finally estimating the obtained characteristic value lambda1,λ2,λ3Is as a1,s2,s3Polynomial λ of coefficient3-s1λ2+s2λ-s3A solution of 0.
Step 6: comparing the network synchronization domain obtained in the step 3 with the characteristic value spectrum estimated in the step 5, and determining the condition for enabling the network to achieve synchronization stability;
utilizing the synchronous domain SR obtained in the step 3 and the eigenvalue estimated value lambda obtained in the step 51,λ2,λ3Let c λ be1,cλ2,cλ3All fall into the range of the synchronous domain SR, and the stability of the NW small-world network in a synchronous state is ensured.
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in:
the invention provides a synchronous analysis method of a NW (NW) small-world network system based on spectral moments, which is different from the existing research on the problem of the complicated network containment synchronization, is based on a random matrix theory, and is used for researching the containment synchronization stability problem of the complicated network from the dynamic evolution angle of the NW small-world network, and the obtained conclusion can better reveal the internal mechanism principle of the mutual influence of the structure evolution and the dynamic behavior of the actual network system, thereby providing theoretical and method guidance for designing the actual network system meeting expected conditions (such as stability and robustness), predicting the behavior of the network system and improving various performances of the actual network system by a control means, and having important economic and social meanings.
Drawings
Fig. 1 is a flow chart of the synchronous analysis method of the NW small-world network system based on the spectrum moment of the present invention;
FIG. 2 is a comparison graph of eigenvalue distribution of matrix C and triangular distribution function of fitted eigenvalue distribution according to the embodiment of the present invention.
Detailed Description
The following detailed description of embodiments of the invention refers to the accompanying drawings.
A method for analyzing synchronization of an NW small-world network system based on spectral moments, as shown in fig. 1, includes the following steps:
step 1: constructing an NW small-world network with N nodes according to an NW small-world network model construction algorithm;
in the NW small-world network, each node represents a coupled oscillator, a network is formed by connecting edges among the nodes, each connecting edge represents the interaction relation among the connected coupled oscillators, the network has a symmetrical connection structure and can be described by a directionless (unweighted) diagram, and the network is adjacent to a matrix
Is a symmetric matrix, the adjacency matrix a may be expressed as follows:
where N represents the number of network nodes,
representing a set of NxN-dimensional real matrices, aijRepresenting the adjacency matrix element, a if there is an edge connection between node i and node jij＝aji1(i ≠ j); otherwise aij＝aji＝0(i≠j)；diDegree representing node i, denoted
D denotes the degree matrix, denoted D ═ diag (D)i)；
Represents a Laplacian matrix, lijIs represented by LaA plain matrix element is marked as L-D-A;
step 2: determining a dynamic equation expression of a node in an NW small-world 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:
wherein,
the state variable representing the ith node, the first derivative with respect to time t being
f (-) represents a single node self dynamic function;
the node state variables are coupled in an inner coupling matrix; constant c>0 represents the global coupling strength;
and step 3: analyzing the synchronous stability of the NW small-world network by using a main stability function method, and determining a synchronous stable region of the NW small-world network system;
synchronization stability of the NW small world network, when x1(t)＝x2(t)＝...＝xN(t) s (t), which indicates that the NW small-world network achieves synchronization stability, where s (t) is called a synchronization state; with the introduction of the containment control, the dynamical equations of the controlled NW small-world network are expressed as:
wherein:
ui(t)＝-cfiΓ(xi(t)-s(t)),i＝1,2,...,N (3)
firepresenting the holdback control gain of the controlled node i，δiIndicating whether the ith node is controlled or not, if the pinning control is applied to the node i, deltai＝1，fiIs greater than 0; else δi＝fi＝0；F＝diag(fi) Representing a holddown control gain matrix; l represents the number of holdback control nodes, and is recorded as
The synchronization state s (t) being a single node system
A solution of (a) satisfies
When the NW small-world network does not reach the synchronous stability, the state of each node has an error with the synchronous state; defining an error vector as εi(t), then xi(t)＝s(t)+εi(t); linearizing and replacing variables of the formula (2)
InRepresenting an n x n dimensional identity matrix,
obtaining:
wherein, Jf(t) is the Jacobian matrix of f (x (t)) in the sync state s (t); lambda [ alpha ]iIs the eigenvalue of matrix C ═ L + F; 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 main stability function method, the region where the maximum Lyapunov exponent of equation (4) is negative is the synchronous stability region of the NW small-world network system, and is denoted as SR ═ c λi|Lmax(cλi)＜0}；
And 4, step 4: based on a spectral moment analysis method, calculating the expected values of the first three spectral moments of the network matrix C after the containment control is introduced according to the probability distribution of the network node degrees, and establishing the relation between the expected spectral moments of the first three and network structure parameters, wherein the network structure parameters comprise the number N of network nodes, the probability p of connecting edges, the containment control gain f and the number l of containment control nodes;
the network matrix C is a real symmetric matrix, { lambda }i1, 2.. N } is recorded as the eigenvalue set of matrix C, then the q-order spectral moments of matrix C are represented as:
when q is equal to 1, the reaction is carried out,
when q is less than or equal to 3, the matrix trace satisfies the switching law, i.e. tr (llf) ═ tr (lfl) ═ tr (fll), and then:
when q is 2, the process is repeated,
when q is 3, the process is repeated,
since the matrix L is D-A, will (D-A)qThe second term is developed and finished to obtain:
wherein T is the total number of triangles contained in the network.
The NW small-world network model considered in this embodiment is to start with a rule graph and to start with a nearest neighbor coupling network having N nodes, where each node is connected to each of k nodes adjacent to the node on the left and right sides; and then, randomizing and adding connecting edges to avoid the occurrence of self-loops and multiple edges, adding an edge on a pair of randomly selected nodes according to the probability p as r/N, wherein at most one edge exists between any two different nodes, and each node cannot have an edge connected with the node. Since random edge adding has little influence on the number of triangles in the network, the method has the advantages of low cost and high efficiency
The node degree distribution of the NW small-world network follows Poisson distribution, i.e.
Wherein d isiRepresenting the degree of a node i in the NW small-world network, d represents a node value, and a parameter r is equal to pN;
calculating the expected value of the first third-order origin moment of the node degree to obtain:
assuming that the first l nodes in the NW small-world network are subjected to containment control, and all control gains are constants f, the expression of the expected value of the first third moment of spectrum of the matrix C at this time is:
for better illustration, in this embodiment, it is assumed that the number N of nodes in the NW small-world network is 512, the number k of adjacent nodes connected to each node in the network is 3, and the probability p of network randomization and edge adding is 0.078, that is, the parameter r is 4. Meanwhile, the first l nodes in the network are selected as the control objects, wherein l is assumed to be 5, and the node degrees are respectively 8, 9, 9, 12 and 7; and the pinning control strengths f are the same, assuming that f is 5. The first third order spectral moment value result of the NW small-world network matrix C obtained by calculation according to the formula (7) and the first third order spectral moment value of the NW small-world network matrix C obtained by actual simulation are shown in the following table:
and 5: constructing a characteristic value spectrum of the piecewise linear function fitting matrix C by using the expected value of the spectrum moment obtained in the step 4;
building the distribution of a piecewise function linear fitting characteristic spectrum by adopting a triangular distribution function according to the expected value of the first third moment of spectrum obtained in the step 4;
knowing the expected value of the first third moment of spectrum, denoted m, of the matrix C obtained in step 41,m2,m3(ii) a To estimate the 3 eigenvalues λ of the matrix C1＜λ2＜λ3Giving m1,m2,m3About lambda1,λ2,λ3The expression of (a) is as follows:
definition of the relation lambda1,λ2,λ3A set of elementary symmetric polynomials sqThe following were used:
s1(λ1,λ2,λ3)＝λ1+λ2+λ3,
s2(λ1,λ2,λ3)＝λ1λ2+λ1λ3+λ2λ3,
s3(λ1,λ2,λ3)＝λ1λ2λ3. (9)
substituting formula (9) for formula (8) to obtain:
by known m1,m2,m3Can be calculated to obtain s1,s2,s3Finally estimating the obtained characteristic value lambda1,λ2,λ3Is as a1,s2,s3Polynomial λ of coefficient3-s1λ2+s2λ-s3A solution of 0. Eigenvalue λ calculated by equation (10)1,λ2,λ3The estimates of (d) are 0.1306, 11.5915, 18.4244, respectively. The distribution of eigenvalues of the matrix C and the triangular distribution function constructed by the eigenvalue estimation values are shown in fig. 2.
Step 6: comparing the network synchronization domain obtained in the step 3 with the characteristic value spectrum estimated in the step 5, and determining the condition for enabling the network to achieve synchronization stability;
utilizing the synchronous domain SR obtained in the step 3 and the eigenvalue estimated value lambda obtained in the step 51,λ2,λ3Let c λ be1,cλ2,cλ3All fall into the range of the synchronous domain SR, and the stability of the NW small-world network in a synchronous state can be ensured.
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 (5)
1. A synchronous analysis method of an NW small-world network system based on spectral moments is characterized in that: the method comprises the following steps:
step 1: constructing an NW small-world network with N nodes according to an NW small-world network model construction algorithm;
in the NW small-world network, each node represents a coupled oscillator, a network is formed by connecting edges among the nodes, each connecting edge represents the interaction relation among the connected coupled oscillators, and the network has a symmetrical connection structure and is adjacent to a matrix
Is a symmetric array, N represents the number of network nodes,
representing a set of NxN-dimensional real matrices, aijRepresenting the adjacency matrix element, a if there is an edge connection between node i and node jij＝aji1(i ≠ j); otherwise aij＝aji＝0(i≠j)；diDegree representing node i, denoted
D denotes the degree matrix, denoted D ═ diag (D)i)；
Represents 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 an NW small-world 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:
wherein,
the state variable representing the ith node, the first derivative with respect to time t being
f (-) represents a single node self dynamic function;
the node state variables are coupled in an inner coupling matrix; constant c>0 represents the global coupling strength;
and step 3: analyzing the synchronous stability of the NW small-world network by using a main stability function method, and determining a synchronous stable region of the NW small-world network system;
and 4, step 4: based on a spectral moment analysis method, calculating the expected values of the first three spectral moments of the network matrix C after the containment control is introduced according to the probability distribution of the network node degrees, and establishing the relation between the expected spectral moments of the first three and network structure parameters, wherein the network structure parameters comprise the number N of network nodes, the probability p of connecting edges, the containment control gain f and the number l of containment control nodes;
and 5: constructing a characteristic value spectrum of the piecewise linear function fitting matrix C by using the expected value of the spectrum moment obtained in the step 4;
building the distribution of a piecewise function linear fitting characteristic spectrum by adopting a triangular distribution function according to the expected value of the first third moment of spectrum obtained in the step 4;
step 6: and (5) comparing the network synchronization domain obtained in the step (3) with the characteristic value spectrum estimated in the step (5) to determine the condition for enabling the network to achieve synchronization stability.
2. The method as claimed in claim 1, wherein the synchronization stability of the NW small-world network in step 3 is when x is1(t)＝x2(t)＝...＝xN(t) s (t), which indicates that the NW small-world network achieves synchronization stability, where s (t) is called a synchronization state; with the introduction of the containment control, the dynamical equations of the controlled NW small-world network are expressed as:
wherein:
ui(t)＝-cfiΓ(xi(t)-s(t)),i＝1,2,...,N (3)
firepresenting 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，fiIs greater than 0; else δi＝fi＝0；F＝diag(fi) Representing a holddown control gain matrix; l represents the number of holdback control nodes, and is recorded as
The synchronization state s (t) being a single node system
A solution of (a) satisfies
When the NW small-world network does not reach the synchronous stability, the state of each node has an error with the synchronous state; defining an error vector as εi(t), then xi(t)＝s(t)+εi(t); linearizing and replacing variables of the formula (2)
InRepresenting an n x n dimensional identity matrix,
obtaining:
wherein, Jf(t) is the Jacobian matrix of f (x (t)) in the sync state s (t); lambda [ alpha ]iIs the eigenvalue of matrix C ═ L + F; p is an approximate equivalent transformation matrix of the matrix CThe approximate form of the matrix C is denoted as Q ═ P-1CP；
According to the main stability function method, the region where the maximum Lyapunov exponent of equation (4) is negative is the synchronous stability region of the NW small-world network system, and is denoted as SR ═ c λi|Lmax(cλi)＜0}。
3. The method as claimed in claim 1, wherein the network matrix C in step 4 is a real symmetric matrix, { λ } λi1, 2.. N } is recorded as the eigenvalue set of matrix C, then the q-order spectral moments of matrix C are represented as:
the initial time network is a nearest neighbor coupling network consisting of N nodes, and the degree of each node is 2 k; random edge connection (avoiding self-loop and multiple edges) is carried out on the basis of the network by using probability p, so that an NW small-world network is obtained, and the node degree distribution of the NW small-world network is subjected to Poisson distribution, namely
Wherein d isiRepresenting the degree of a node i in the NW small-world network, d represents a node value, and a parameter r is equal to pN;
assuming that the first l nodes in the NW small-world network are subjected to containment control, and all control gains are constants f, the expression of the expected value of the first third moment of spectrum of the matrix C at this time is:
4. the method as claimed in claim 1, wherein the expected value of the first third moment in step 5 is denoted as m1,m2,m3(ii) a To estimate the 3 eigenvalues λ of the matrix C1＜λ2＜λ3Giving m1,m2,m3About lambda1,λ2,λ3The expression of (a) is as follows:
definition of the relation lambda1,λ2,λ3A set of elementary symmetric polynomials sqThe following were used:
s1(λ1,λ2,λ3)＝λ1+λ2+λ3,
s2(λ1,λ2,λ3)＝λ1λ2+λ1λ3+λ2λ3,
s3(λ1,λ2,λ3)＝λ1λ2λ3. (9)
substituting formula (9) for formula (8) to obtain:
s1＝3m1,
by known m1,m2,m3Is calculated to obtain s1,s2,s3Finally estimating the obtained characteristic value lambda1,λ2,λ3Is as a1,s2,s3Polynomial λ of coefficient3-s1λ2+s2λ-s3A solution of 0.
5. The method as claimed in claim 1, wherein the comparing step 6 comprises comparing the network synchronization domain obtained in step 3 with the eigenvalue spectrum estimated in step 5, wherein the synchronization domain SR obtained in step 3 and the eigenvalue estimation value λ obtained in step 5 are used1,λ2,λ3Let c λ be1,cλ2,cλ3All fall into the range of the synchronous domain SR, and the stability of the NW small-world network in a synchronous state is ensured.
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| CN113467242A (en) * | 2021-07-06 | 2021-10-01 | 东北大学秦皇岛分校 | Method for analyzing divergence of synchronous domain of time-lag coupling network system under control of constraint |
| CN116345430A (en) * | 2022-11-24 | 2023-06-27 | 兰州理工大学 | Synchronous oscillation finite time function projection control method for micro-grid |
| CN116582448A (en) * | 2023-07-13 | 2023-08-11 | 三峡科技有限责任公司 | Network optimization method based on elastic channel network index |
Citations (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103870691A (en) * | 2014-03-13 | 2014-06-18 | 西南交通大学 | Building method of complex network |
| CN107393522A (en) * | 2016-05-11 | 2017-11-24 | 哈曼贝克自动系统股份有限公司 | Select the method and system of the sensing station of active road noise control on vehicle |
| CN108732926A (en) * | 2018-06-05 | 2018-11-02 | 东北石油大学 | Networked system method for estimating state based on insufficient information |
| CN109818792A (en) * | 2019-01-29 | 2019-05-28 | 西安电子科技大学 | A kind of second order complex dynamic network synchronisation control means containing oriented time-varying topology |
| CN110543695A (en) * | 2019-08-14 | 2019-12-06 | 天津大学 | electric-gas coupling comprehensive energy system feasible region calculation method |
| CN111884849A (en) * | 2020-07-23 | 2020-11-03 | 东北大学秦皇岛分校 | Random network system containment synchronization stability analysis method based on spectrum moment |
| CN113792710A (en) * | 2021-11-15 | 2021-12-14 | 滨州学院 | Spectrum reconstruction method and device and electronic equipment |
-
2020
- 2020-10-09 CN CN202011072649.6A patent/CN112464414A/en active Pending
Patent Citations (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103870691A (en) * | 2014-03-13 | 2014-06-18 | 西南交通大学 | Building method of complex network |
| CN107393522A (en) * | 2016-05-11 | 2017-11-24 | 哈曼贝克自动系统股份有限公司 | Select the method and system of the sensing station of active road noise control on vehicle |
| CN108732926A (en) * | 2018-06-05 | 2018-11-02 | 东北石油大学 | Networked system method for estimating state based on insufficient information |
| CN109818792A (en) * | 2019-01-29 | 2019-05-28 | 西安电子科技大学 | A kind of second order complex dynamic network synchronisation control means containing oriented time-varying topology |
| CN110543695A (en) * | 2019-08-14 | 2019-12-06 | 天津大学 | electric-gas coupling comprehensive energy system feasible region calculation method |
| CN111884849A (en) * | 2020-07-23 | 2020-11-03 | 东北大学秦皇岛分校 | Random network system containment synchronization stability analysis method based on spectrum moment |
| CN113792710A (en) * | 2021-11-15 | 2021-12-14 | 滨州学院 | Spectrum reconstruction method and device and electronic equipment |
Non-Patent Citations (2)
| Title |
|---|
| VICTOR M.PRECIADO等: "《Moment-Based Analysis of Synchronization in Small-World Networksof Oscillators》", 《ARXIV》 * |
| 项林英等: "《复杂动态网络的建模、分析与控制研究综述》", 《自然科学进展》 * |
Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113467242A (en) * | 2021-07-06 | 2021-10-01 | 东北大学秦皇岛分校 | Method for analyzing divergence of synchronous domain of time-lag coupling network system under control of constraint |
| CN116345430A (en) * | 2022-11-24 | 2023-06-27 | 兰州理工大学 | Synchronous oscillation finite time function projection control method for micro-grid |
| CN116345430B (en) * | 2022-11-24 | 2023-10-10 | 兰州理工大学 | Synchronous oscillation finite time function projection control method for micro-grid |
| CN116582448A (en) * | 2023-07-13 | 2023-08-11 | 三峡科技有限责任公司 | Network optimization method based on elastic channel network index |
| CN116582448B (en) * | 2023-07-13 | 2023-12-01 | 三峡科技有限责任公司 | Network optimization method based on elastic channel network index |
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