CN112363388A - Complex network node dynamic classification control method based on connection relation observer - Google Patents

Complex network node dynamic classification control method based on connection relation observer Download PDF

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CN112363388A
CN112363388A CN202011222682.2A CN202011222682A CN112363388A CN 112363388 A CN112363388 A CN 112363388A CN 202011222682 A CN202011222682 A CN 202011222682A CN 112363388 A CN112363388 A CN 112363388A
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subsystem
observer
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CN112363388B (en
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王颀
熊建斌
岑健
胡俊敏
陈康
刘鸣慧
钱文博
刘文浩
李春林
余得正
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Guangdong Polytechnic Normal University
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Abstract

The invention discloses a complex network node dynamic classification control method based on a connection relation observer, which comprises the following steps: s1, aiming at the undirected complex network, the rationality of using a Riccati matrix differential equation as a connection relation subsystem model is proved; s2, designing a special coupling item form of the connection relation subsystem to couple the connection relation subsystem and the node subsystem in the complex network; s3, designing a state observer of the connection relation aiming at the connection relation subsystem; and S4, designing controllers aiming at the connection relation subsystem and the node subsystem by using the information in the connection relation subsystem observer, so that the nodes of the complex network realize dynamic classification. The invention uses the information in the observer to design the controller aiming at the connection relation subsystem, so that the complex network asymptotically tracks a known classifiable network, thereby achieving the purpose of dynamically classifying the complex network.

Description

Complex network node dynamic classification control method based on connection relation observer
Technical Field
The invention belongs to the technical field of automatic control, and particularly relates to a complex network node dynamic classification control method based on a connection relation observer.
Background
In real life, a plurality of dynamic complex systems exist, and from the viewpoint of mathematical graph theory, the systems can be abstracted into a dynamic complex network for description. Similar to many complex systems in reality, the nodes of a complex network may also be divided into a number of categories. Therefore, in the past years, the classification of complex network nodes becomes a hot topic and a great deal of research results emerge. These algorithms are all divided by the density of the connection relationships between nodes.
However, these algorithms described above do not take into account the sign of the connection relationships and are therefore not applicable to a sign network. In real life, a plurality of social networks can be abstracted into a symbolic network, and the research on the node classification method applicable to the symbolic network has important significance. In the existing related research results, a class of algorithms is developed from the traditional unsigned network node classification algorithm, such as the FEC algorithm, the Laplacian algorithm, and the like. However, these algorithms do not provide accurate classification results because they do not take full advantage of the information provided by negative connections in the network, although they take into account negative connections. Amelio et al propose a genetic algorithm which takes the weight sum of negative connection relations in the same class as a target function and achieves the purpose of symbol network node classification by continuously optimizing the function; jiang et al propose an SSBM model to measure the congestion of a network as a statistical probability model whose parameters reflect the probability that nodes belong to different classes and the centrality of each node in its class. These algorithms classify nodes according to the density of the connection relations, which means that these algorithms cannot strictly ensure that the connection relations between nodes in the same class are positive, and the connection relations between nodes in different classes are non-positive. In other words, these algorithms do not recognize the important role of connection relation notation in node classification.
In fact, the sign of the connection plays an important role in the sign network. For example, in social networks, friendly, cooperative relationships between individuals are typically represented by positive connection relationships, and hostile, antagonistic relationships between individuals are typically represented by negative connection relationships. In a neural network, a positive connection relationship and a negative connection relationship between neurons represent a mutual promotion and inhibition between neurons, respectively. Therefore, in the symbolic network, it is important to study how to classify nodes by the symbols of the connection relationships.
Some studies have been conducted by some scholars. For example, Wang et al investigated the sufficient requirements for the classifiable nature of generalized symbolic network nodes based on the concept of structural holes and hole owners (brokers). The document states that when the structure of the hole owner (Broker) is not present in the generalized symbol network, the generalized symbol network can be divided into several categories. However, the above algorithm is proposed for a static symbol network and is not suitable for a dynamic network.
For dynamic symbolic networks, Chen et al propose a DEC algorithm based on node phase, the core idea of which is to gradually bring together two nodes connected by a positive connection relation and gradually separate two nodes connected by a negative connection relation by a change in node phase. However, this method is only applied to the case where the node state is a one-dimensional variable, and if the node state is multidimensional, the above method is not applicable. Gao et al propose a node-based adaptive complex network control strategy, which ultimately divides the nodes of the network into two categories and cannot classify the nodes of the symbolic network into multiple categories.
Therefore, aiming at the defects of the research results, the invention provides a complex network node dynamic classification control method based on a connection relation observer. First, a complex network is considered as a large system formed by coupling a node subsystem and a connection relation subsystem. By designing a state observer for the state (connection relation weight) of the connection relation subsystem and designing a controller for the connection relation subsystem by using information in the observer, the complex network gradually tracks a known classifiable network, thereby achieving the purpose of dynamic classification of the complex network.
Disclosure of Invention
The invention aims to solve the defects in the prior art, and provides a complex network node dynamic classification control method based on a connection relation observer.
In order to achieve the purpose, the invention provides the following technical scheme:
a complex network node dynamic classification control method based on a connection relation observer comprises the following steps:
s1, aiming at the undirected complex network, it is proved that the Riccati matrix differential equation can be used as the result of the approximate linearization of the undirected complex network under certain assumed conditions, and therefore, the rationality of using the Riccati matrix differential equation as a connection relation subsystem model for the undirected network is proved;
s2, designing a special coupling item form of the connection relation subsystem to couple the connection relation subsystem and the node subsystem in the complex network;
s3, designing a state observer of the connection relation aiming at the connection relation subsystem;
and S4, designing controllers aiming at the connection relation subsystem and the node subsystem by using the information in the connection relation subsystem observer, so that the nodes of the complex network realize dynamic classification.
Preferably, in the step S1, for the undirected complex dynamic network, through a series of assumed conditions, it is proved that the Riccati matrix differential equation is a result of approximately linearizing the dynamic equation, and further, the rationality of using the Riccati matrix differential equation as the connection relation subsystem model is explained.
Preferably, the step S2 designs the following shapes: Φ (z) ═ Λ (z)+Λ(z)TThe coupling term of the connection relation subsystem in the formula, wherein Λ (z) ═ J-1ΓTζ(z)J-1And | | | ζ (z) | | is less than or equal to η (t) | | z | | non-woven phosphor2，η(t)＞0。
Preferably, the step S3 designs the following shapes:
Figure BDA0002762595550000031
connection relation subsystem state observer in formula
Figure BDA0002762595550000032
And the robust term in the formula is used for observing the unknown connection relation state.
Preferably, in step S4, by using the information of the state observer of the connection relation subsystem, the shape of the observer is designed as follows:
Figure BDA0002762595550000033
formula specific node subsystem and
Figure BDA0002762595550000041
and the formula aims at controlling the connection relation subsystem to realize the dynamic classification of the complex network nodes.
Compared with the prior art, the invention has the beneficial effects that: compared with the prior art, the complex network node dynamic classification control method based on the connection relation observer provided by the invention takes the complex network as a large system formed by coupling a node subsystem and a connection relation subsystem. By designing a state observer for the state (connection relation weight) of the connection relation subsystem and designing a controller for the connection relation subsystem by using information in the observer, the complex network gradually tracks a known classifiable network, thereby achieving the purpose of dynamic classification of the complex network.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
A complex network node dynamic classification control method based on a connection relation observer comprises the following steps:
s1, aiming at the undirected complex network, it is proved that the Riccati matrix differential equation can be used as the result of the approximate linearization of the undirected complex network under certain assumed conditions, and therefore, the rationality of using the Riccati matrix differential equation as a connection relation subsystem model for the undirected network is proved;
s2, designing a special coupling item form of the connection relation subsystem to couple the connection relation subsystem and the node subsystem in the complex network;
s3, designing a state observer of the connection relation aiming at the connection relation subsystem;
and S4, designing controllers aiming at the connection relation subsystem and the node subsystem by using the information in the connection relation subsystem observer, so that the nodes of the complex network realize dynamic classification.
Specifically, in the step S1, for the undirected complex dynamic network, through a series of assumed conditions, it is proved that the Riccati matrix differential equation is a result of approximately linearizing the dynamic equation, and the rationality of using the Riccati matrix differential equation as the connection relation subsystem model is further explained.
Specifically, the step S2 is designed to have the following shapes: phi (z) ═ Λ (z) + Λ (z)TThe coupling term of the connection relation subsystem in the formula, wherein Λ (z) ═ J-1ΓTζ(z)J-1And | | | ζ (z) | | is less than or equal to η (t) | | z | | non-woven phosphor2，η(t)＞0。
Specifically, the step S3 is designed to have the following shapes:
Figure BDA0002762595550000051
connection relation subsystem state observer in formula
Figure BDA0002762595550000052
And the robust item in the formula is used for observing the unknown connection relation state (connection relation weight).
Specifically, in step S4, by using the information of the state observer of the connection relationship subsystem, the shape of the observer is designed as follows:
Figure BDA0002762595550000053
formula specific node subsystem and
Figure BDA0002762595550000054
and the formula aims at controlling the connection relation subsystem to realize the dynamic classification of the complex network nodes.
Considering a time-varying undirected complex dynamic network composed of N nodes with full real connections, when the coupling relationship between a node and its connecting line is not considered, the dynamic equation of an isolated connection relationship subsystem can be generally expressed as:
Figure BDA0002762595550000055
wherein xij(t) represents the connection relationship between the node i and the node j at the time t; x ═ Xij)∈RN×NState matrix representing the connection relation subsystem, fij(X) is with respect to XijN of (A)2A primitive smoothing function, i, j ═ 1,2, …, N, satisfying fij(X)＝fji(X)。
Definitions 1 consider equation (1) if a matrix of constants exists
Figure BDA0002762595550000061
Satisfying f for any i, j e {1,2, …, N }ij(X*) Is not identical to 0, then X is called X ═ X*Is a balanced matrix of the connection relation subsystem (1).
Notation F (X) ═ fij(X))N×N. The dynamic equation (1) can be written in the form of a matrix as follows:
Figure BDA0002762595550000062
thus, as can be seen from definition 1 and equation (2) for dynamics, if and only if F (X)*) X ≡ 0 or X ═ X*Is a balance matrix of the network connection relation subsystem (2).
Mapping matrix X into vector space by a straightening operation, then define the balanced matrix X ═ X in 1*Can be regarded as the equilibrium point (state) of the dynamic equation (2) in the Lyapunov meaning, and notes the Euclidean norm | | | X-X*||＝||vec(X)-vec(X*) I, so for the balancing matrix X*In other words, the concepts of stability, asymptotic stability and the like in the Lyapunov sense can be naturally extended to the equation (2) of the dynamic state.
Definition 2 if equilibrium matrix X of equation (2) is X*And the network connection relation subsystem expressed by the equation (2) is stable (asymptotically stable) if the network connection relation subsystem is stable (asymptotically stable) in the Lyapunov sense.
Considering a network connection relation subsystem (2) with a balancing matrix of
Figure BDA0002762595550000063
Obviously, the dynamic equation (2) can be converted into a differential equation in the form of a vector by using matrix straightening mapping
Figure BDA0002762595550000064
Therefore, the approximate linearization method in the Lyapunov stability theory can be utilized to discuss the differential equation at the equilibrium point vec (X)*) The problem of approximate linearization. However, such an approximate linearization result has two drawbacks, namely, the state dimension of the approximate linearization system is increased to N2For a complex network with many nodes, the amount of calculation increases sharply, and the intuitiveness of the connection relationship between the network nodes is destroyed (obviously vec (X) is not as intuitive as X). Therefore, we will discuss the dynamic equation (2) in its balancing matrix based on some characteristics of the connection relationship between the network nodes
Figure BDA0002762595550000065
OfThe problem of linearization.
The following research results show that under certain conditions, the approximate linear dynamic equation of the complex network connection relation subsystem is a Riccati type matrix differential equation.
Considering equation (1) of motion, note that fij(X)＝fji(X), we need the following assumptions.
Assume 1 considers the dynamic equation (1), fij(X) can be represented as follows:
fij(X)＝δij(xi1 xi2 … xiN)+δji(xj1 xj2 … xjN) (3)
wherein the smoothing function deltaij(·)＝δji(. a) and satisfy
Figure BDA0002762595550000071
(a) Suppose equation (3) in 1 states that for a given i, j, the function fij(X) is with respect to X onlyik、xjk(k is 1,2, …, N) (the number of arguments is at most 2N, not N)2One). (b) Due to xijRepresents the weight value of the connection relationship between the ith node and the jth node, so that the assumption that equation (3) in 1 means that only the "connection relationship x related to the ith node in the networkik(k ═ 1,2, …, N) "and" connection relationship x associated with the jth nodejk(k ═ 1,2, …, N) "influences xijRate of change of
Figure BDA0002762595550000072
If assume 1 holds, then δij(·)＝δjiThus it is easy to see
Figure BDA0002762595550000073
Note the book
Figure BDA0002762595550000074
Figure BDA0002762595550000075
Then there is
Figure BDA0002762595550000076
By Taylor's formula, the function (3) is
Figure BDA0002762595550000077
The process can be developed to obtain:
Figure BDA0002762595550000078
Figure BDA0002762595550000081
wherein,
Figure BDA0002762595550000082
respectively show about
Figure BDA0002762595550000083
Figure BDA0002762595550000084
The higher order of (a) is infinitesimally small.
In Taylor equation (4), the higher order terms are ignored
Figure BDA0002762595550000085
The following approximate equation can be obtained:
Figure BDA0002762595550000086
consider the following N × N order real matrices:
Figure BDA0002762595550000087
memory matrix
Figure BDA0002762595550000088
Wherein
Figure BDA0002762595550000089
If it is assumed that 1 is true, then using equations (5) and (6) we can get dynamic equation (2) where the balance matrix X is X*First approximation of (d):
Figure BDA00027625955500000810
suppose that the N matrices in equation (6) of 2 are all equal, i.e., there is a constant matrix P that satisfies Pk＝P，k＝1,2,…,N。
It can be easily seen that the existence of an N-ary real function δ (-) makes the smooth function δ in hypothesis 1ij(·)＝δjiδ (·), then assume 5.6 holds, which means that the connection relation x in the network is trueik、xjk(k ═ 1,2, …, N) for xijRate of change
Figure BDA0002762595550000091
The manner of influence is the same.
Easy to verify identity matrix
Figure BDA0002762595550000092
Thus, when assuming 1-2 is true, the dynamic equation (7) can be simplified as:
Figure BDA0002762595550000093
namely:
Figure BDA0002762595550000094
the dynamic equation (8) is a matrix differential equation of the Riccati type, which also explains the rationality of describing dynamic changes of the social network connection relationship using the Riccati differential equation from another point of view.
In this context, consider a generalized symbolic network comprising N nodes, the dynamic equation for each node i satisfying the following equation:
Figure BDA0002762595550000095
wherein z isi＝[zi1 zi2 … zin]T∈RnRepresenting the node state vector of the ith node. A. thei∈Rn×n；fi(zi)＝[fi1(zi1) fi2(zi2) … fin(zin)]T∈RnRepresenting a continuous non-linear function vector. Gj(zj)＝[Gj1(zj1) Gj2(zj2) … Gjn(zjn)]TRepresenting an in-coupling nonlinear function vector. c. CiIndicating the coupling strength. u. ofiRepresenting the control action exerted on node i. x is the number ofijRepresenting the weight of the connection relationship between the node i and the node j.
Introducing vector z ═ z1 T z2 T … zN T]T；A＝diag(A1 A2 … AN)；f(z)＝[f1(z1)T f2(z2)T… fN(zN)T]T∈RNn；c＝diag(c1Π c2Π … cNΠ)，Π∈RnA full one vector representing n dimensions; x ═ Xij(t))N×N；G(z)＝[G1(z1)T G2(z2)T … GN(zN)T]T∈RNn；u＝[u1 T u2 T … uN T]T。
In conjunction with the definition of the Kronecker product, node subsystem dynamic equation (9) can be rewritten as:
Figure BDA0002762595550000101
consider that the connection relation subsystem satisfies the following Riccati matrix differential equation:
Figure BDA0002762595550000102
wherein P ∈ RN×NIs a matrix of real numbers, theta is a known matrix of constants, phi (z) is e RN×NIs the coupling relation between the connection relation subsystem and the node subsystem, and U is the control input of the connection relation subsystem. Y is formed by the element Rp×NIs the state output of the connection relation subsystem, and is epsilon Rp×NIs the output gain matrix.
Depending on the straightening operation and the nature of the Kronecker product, equation (11) can be rewritten as:
Figure BDA0002762595550000103
wherein,
Figure BDA0002762595550000104
representing an identity matrix of order N.
Suppose 3 assumes that (P, Γ) is fully stable for the connection relation subsystem (11). That is, there is a matrix W ∈ RN×pSuch that P + wtz is a Hurwitz matrix.
If hypothesis 3 holds, then there is a positive definite matrix J ∈ RN×NSuch that for any matrix Q > 0, the following Lyapunov equation is satisfied:
(P+WΓ)TJ+J(P+WΓ)＝-Q (13)
lemma 1 if it is assumed that the equations 3 and (13) hold, the following equation holds:
Figure BDA0002762595550000105
Figure BDA0002762595550000111
wherein,
Figure BDA0002762595550000112
and (3) proving that: from the formula (13), we can obtain:
Figure BDA0002762595550000113
Figure BDA0002762595550000114
by using the properties of the Kronecker product, it is known from the formulae (14) and (15):
Figure BDA0002762595550000115
Figure BDA0002762595550000116
multiplying both the left and right sides of the formulas (16a) and (16b) by
Figure BDA0002762595550000117
And
Figure BDA0002762595550000118
and notice that
Figure BDA0002762595550000119
It is possible to obtain:
Figure BDA00027625955500001110
Figure BDA00027625955500001111
the theory of leading 1 can be used for the evidence.
Assume 4 that Φ (z) in the connection relation subsystem (11) satisfies the following equation:
Φ(z)＝Λ(z)+Λ(z)T (18)
wherein Λ (z) ═ J-1ΓTζ(z)J-1And | | | ζ (z) | | is less than or equal to η (t) | | z | | non-woven phosphor2，η(t)＞0。
Considering that the state of the link relation subsystem is not measurable, before controller design, the following state observer is first designed to estimate the state of the link relation subsystem:
Figure BDA00027625955500001112
wherein,
Figure BDA0002762595550000121
and representing the estimated value of the state matrix X of the connection relation subsystem at the time t.
Figure BDA0002762595550000122
Which represents the output of the state observer,
Figure BDA0002762595550000123
a robust term is represented.
In observer system (5.52) form, robust term
Figure BDA0002762595550000124
Satisfies the following conditions:
Figure BDA0002762595550000125
wherein,
Figure BDA0002762595550000126
using the Kronecker product and the correlation properties of the straightening operation, one can obtain:
Figure BDA0002762595550000127
lemma 2 if assumption 3 holds, the error between the estimated state in (19) and the state of the connection relation subsystem
Figure BDA0002762595550000128
Is asymptotically stable.
And (3) proving that: by using the formulae (10) and (19), it is possible to obtain:
Figure BDA0002762595550000129
selecting positive definite functions
Figure BDA00027625955500001210
Its orbital derivative is:
Figure BDA00027625955500001211
Figure BDA0002762595550000131
as can be seen from (23), the estimation error E is bounded, and
Figure BDA0002762595550000132
refer to 2 for evidence.
A control target: suppose X*∈RN×NThe network connection relation matrix can be classified for a known node. By using the states in a state observer (19)
Figure BDA0002762595550000133
And the state z (t) of the node subsystem (10), the design control U for the connection relation subsystem (11) and the design control for the node subsystemu such that the state X (t) of the connectivity subsystem tracks the known classifiable network connectivity matrix X*And the state of the node subsystem is guaranteed to be bounded, so that node classification is realized.
In order to achieve the control goal, the following node subsystem controllers u are selected:
Figure BDA0002762595550000141
selecting the following controllers U of the connection relation subsystems:
Figure BDA0002762595550000142
let e be X-X*Then, there are:
Figure BDA0002762595550000143
theorem 1 considers a generalized symbolic network consisting of a node subsystem shown in (10) and a connection relation subsystem shown in (11). A state observer in the form of (17) is built for the connection relation subsystem. If assumptions 3-4 are true, then under the influence of controllers (24) and (25), state X of the connectivity relation subsystem may asymptotically track a given node classifiable network connectivity matrix X*And the state of the node subsystem
Figure BDA0002762595550000144
And (3) proving that: selecting positive definite function
Figure BDA0002762595550000145
Its derivative with time is:
Figure BDA0002762595550000146
as can be seen from the introduction 2,
Figure BDA0002762595550000147
then, according to the formula (27):
Figure BDA0002762595550000148
as can be seen from equation (28), the error in the error system (26)
Figure BDA0002762595550000149
And the state of the node subsystem
Figure BDA00027625955500001410
After the syndrome is confirmed.
In summary, the following steps: compared with the prior art, the complex network node dynamic classification control method based on the connection relation observer provided by the invention takes the complex network as a large system formed by coupling a node subsystem and a connection relation subsystem. By designing a state observer for the state (connection relation weight) of the connection relation subsystem and designing a controller for the connection relation subsystem by using information in the observer, the complex network gradually tracks a known classifiable network, thereby achieving the purpose of dynamic classification of the complex network.
Finally, it should be noted that: although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that modifications may be made to the embodiments or portions thereof without departing from the spirit and scope of the invention.

Claims (5)

1. A complex network node dynamic classification control method based on a connection relation observer is characterized in that: the method comprises the following steps:
s1, aiming at the undirected complex network, it is proved that the Riccati matrix differential equation can be used as the result of the approximate linearization of the undirected complex network under certain assumed conditions, and therefore, the rationality of using the Riccati matrix differential equation as a connection relation subsystem model for the undirected network is proved;
s2, designing a special coupling item form of the connection relation subsystem to couple the connection relation subsystem and the node subsystem in the complex network;
s3, designing a state observer of the connection relation aiming at the connection relation subsystem;
and S4, designing controllers aiming at the connection relation subsystem and the node subsystem by using the information in the connection relation subsystem observer, so that the nodes of the complex network realize dynamic classification.
2. The method for controlling the dynamic classification of the complex network nodes based on the connection relation observer as claimed in claim 1, wherein: in the step S1, for the undirected complex dynamic network, through a series of assumed conditions, it is proved that the Riccati matrix differential equation is a result of approximate linearization of the dynamic equation, and further, the rationality of using the Riccati matrix differential equation as the connection relation subsystem model is explained.
3. The method for controlling the dynamic classification of the complex network nodes based on the connection relation observer as claimed in claim 1, wherein: the step S2 designs the following shapes: phi (z) ═ Λ (z) + Λ (z)TThe coupling term of the connection relation subsystem in the formula, wherein Λ (z) ═ J-1ΓTζ(z)J-1And | | | ζ (z) | | is less than or equal to η (t) | | z | | non-woven phosphor2，η(t)＞0。
4. The method for controlling the dynamic classification of the complex network nodes based on the connection relation observer as claimed in claim 1, wherein: the step S3 designs the following shapes:
Figure FDA0002762595540000011
connection relation subsystem state view in formulaMeasuring device and
Figure FDA0002762595540000012
and the robust term in the formula is used for observing the unknown connection relation state.
5. The method for controlling the dynamic classification of the complex network nodes based on the connection relation observer as claimed in claim 1, wherein: in the step S4, by using the information of the state observer of the connection relation subsystem, the shape of the observer is designed as follows:
Figure FDA0002762595540000021
formula specific node subsystem and
Figure FDA0002762595540000022
and the formula aims at controlling the connection relation subsystem to realize the dynamic classification of the complex network nodes.
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