# Structural Balance Control of Complex Dynamical Networks Based on State Observer for Dynamic Connection Relationships

Authors: Zilin Gao; Yinhe Wang; Jiang Xiong; Yong Pan; Yuanyuan Huang
Journal: Complexity
 (2020)

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## Abstract

This paper investigates the observer-based structural balance control for a class of complex dynamical networks. Generally speaking, a complete complex dynamical network is composed of two coupled subsystems, which are called node subsystem (NS) and connection relationship subsystem (CS), respectively. Similar to synchronization and stabilization of networks, the structural balance is another phenomenon of networks and determined by the state of connection relationships. However, it is not feasible to design the controller for the CS directly because the states of the connection relationships are difficult to be measured accurately in practical applications. In order to solve this problem, a state observer for the CS has been designed. Thus, the structural balance controller in the CS can be directly designed by using the estimation information of the state observer. Then, with the help of the Lyapunov stability theory, it is proved that the CS can asymptotically track a given structural balance matrix under the influence of the observer-based controller. Finally, the results derived from this paper are demonstrated by performing a numerical example.


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## Body

## 1. Introduction

In recent decades, complex dynamical networks have drawn considerable attention because many real-world applications need the help of research results about complex dynamical networks, such as neural networks [1], social networks [2], power networks [3], wireless networks [4], and Internet networks [5]. Complex dynamical networks are represented by a group of dynamically interacting nodes with connection relationships between them. That is, the dynamical equations of a complete network mainly contain two parts. The first is the node subsystem (nodes) and the second is the connection relationship subsystem (connection relationships between nodes), where the two subsystems are usually coupled with each other. Moreover, the model of complex dynamical networks helps to understand and examine the dynamical behaviors of networks in a better way. The reason is that some behaviors are reflected by nodes such as synchronization [6, 7], stabilization [8, 9], and consensus [10–12], and another characteristic behavior is determined by the connection relationships such as structural balance [13–15].From the existing research results about the synchronization, stabilization, consensus, and structural balance or other problems of complex dynamical networks, it is assumed that all states in complex dynamical networks, including the states of nodes and connection relationships, can be measured accurately. In fact, due to the large scale of complex dynamical networks (including a large number of nodes and connection relationships) and influence of external environment, technical constraints, and measurement costs, this assumption is too hard to meet for a network [16]. It implies that it is very necessary and important to construct state observers for the complex dynamical networks to estimate the unknown state variables. In [17, 18], the state estimation problem is investigated for complex dynamical networks with the coupling time delays. In [19, 20], the state estimation problem of complex networks with uncertain network parameters and stochastic noisy disturbance is discussed. The papers [21, 22] proposed some state estimation methods to solve the problem of packet loss caused by the congestion and transmission error in the networks.However, the abovementioned papers about the state estimation of complex dynamical networks only discussed the estimation problems of the states in nodes subsystem and assumed that the connection relationships between nodes are known. On the other words, the measurement and state estimation problems of the connection relationships not only are ignored but also they are not concerned with the structural balance control problem of the complex dynamical networks.Compared to the states of the nodes in the networks, the state values of the connection relationships are more complex and difficult to be measured accurately in practical applications, especially in some real networks such as social networks. Hence, there are only a few papers to study the effective measurement problem of the connection relationships in networks. In [23], a state measurement method of connection relationships is proposed according to the physical interaction between the individuals (nodes). Zeng et al. [24] introduced a measurement method of the social relationships based on adaptive weights. Similar to the state values of the nodes, it is also difficult to obtain all the measured values of the connection relationships, and designing observers to estimate the unmeasured state values of the connection relationships is very necessary. As far as we know, there is only one paper to discuss the state estimation problem of dynamic connection relationships in networks [25]. Unfortunately, this paper only solves the state estimation problem for connection relationships in a class of networks, but does not consider some related control problems, such as structural balance control.On the other hand, more and more scholars pay attention to the structural balance of networks, and a series of research results have been obtained [13–15, 24, 26–34]. So far, the models of structural balance have gone through the following stages. (i) The signed network model, in which the state values of connection strength are expressed by only using integers 0 and ±1 [24, 26]; (ii) the time continuous models, which are described by a set of differential equations or a matrix differential equation [13, 15, 27–32], but they did not consider the effects of internal state motions of the nodes on the evolution of connection relationships; (iii) the model is described by a Riccati dynamical equation with the coupling matrix about the internal state of nodes [14, 33, 34], which implies that any dynamical changes of nodes (or connection relationships) will cause the connection relationships (or nodes) to be changed via the effective coupling. These papers investigate the time evolution behaviors of the connection relationships and show the structural balance of networks. However, considering that the state values of connection relationships cannot be measured accurately; thus, there are only a few papers to discuss the structural balance control problems of networks. In [15, 31], the structural balance of networks is realized by changing the values of the connection relationships between the nodes and a “reference” agent (node). The papers [14, 34] proposed a complex dynamical network model composed of the node subsystem and connection relationship subsystem, where the two subsystems are coupled with each other. Thus, the structural balance of networks can be achieved by designing the controller in node subsystem, which can control the states of nodes to force the connection relationship subsystem to be structural balance via the effective coupling. To the best of our knowledge, the structural balance control of networks based on the state observer of connection relationships has not been studied.Inspired by the above discussions, we mainly focus on the design of the structural balance controller with the estimation information of the state observer for the connection relationships subsystem in this paper. Firstly, we proposed a complex dynamical network model, which is composed of the nodes subsystem and connection relationships subsystem with coupled each other. Then, we designed a state observer for the connection relationship subsystem. Finally, the controller in connection relationships subsystem has been proposed by using the estimation information of the state observer, which can guarantee the phenomenon of structural balance emerges in the network. That is, the estimation and direct control problems of the connection relationships subsystem are solved in this paper.This paper is organized as the following sections. Section2 proposes a complex dynamical network model, which is composed of the nodes subsystem and the connection relationships subsystem with outputs. In Section 3, the state observer for the connection relationships subsystem is designed. Section 4 completes the design of the structural balance controller in connection relationship subsystem by using the estimation information of the state observer. The simulation example is presented in Section 5. Finally, the conclusions are given in Section 6.


## 2. Network Model Description

In this paper, we consider a class of the undirected complex dynamical network withN nodes, which are composed of the node subsystem and the connection relationship subsystem. If each node is n-dimensional continuous-time system, then the node subsystem and the connection relationship subsystem can be described as follows (see [14, 33, 34]):(1)x˙i=fixi+ci∑j=1NpijtHjxj,i=1,2,…,N,(2)P˙=ΘP+PΘT+U+UT+Gx+GxT,Y=ϒP,where xi=xi1xi2…xinT∈Rn is the state vector of node i; the vector functions fixi=fi1xifi2xi…finxiT and Hjxj=Hj1xjHj2xj…HjnxjT; ci is a given positive constant, and it represents the common connection strength of the ith node in the networks; the matrix Θ∈RN×N is a constant matrix; Gx∈RN×N represents the coupling matrix with the internal state of the nodes, and x=x1Tx2T…xNTT∈RNn; U∈RN×N is the control input; ϒ∈RN1×N is the output matrix of the connection relationship subsystem (2); and the connection relationships matrix P=Pt=pijtN×N, where the time-varying function pijt expresses the weighted value of connection relationship between the node i and the node j in the network and pji=pij for undirected networks, especially when i=j, pij indicates the relationship strength of the node itself, i=1,2,…,N.Now, we introduce the definitions ofvec⋅ operator and the Kronecker product of matrices as follows.Definition 1.
(see [35]). The application vec:Rk×l⟶Rkl, defined by(3)vecH=h11,…,h1l,h21,…,h2l,…,hk1,…,hklT,where H=hijk×l∈Rk×l is called the vectorization operator.Definition 2 (see [35]).
IfH∈Rk×landZ∈Rc×d, then the Kronecker product of H and Z, denoted as H⊗Z∈Rkc×ld, is defined by the following matrix:(4)H⊗Z=h11Zh12Z⋯h1lZh21Zh22Z⋯h2lZ⋅⋅⋅⋅⋅⋅⋅⋅hk1Zhk2Z⋯hklZ.
Then, the following basic properties of the vectorization operatorvec⋅ and Kronecker product can be obtained from [35].(1)
H⊗XW⊗D=HW⊗XD(2)
H⊗DT=HT⊗DT(3)
X⊗W−1=X−1⊗W−1(4)
vecHXZ=H⊗ZTvecX(5)
vecHX+XZ=H⊗I+I⊗ZTvecXwhere X and W are matrices with compatible dimensions and I represents the identity matrix with compatible dimensions. Especially, it is assumed that both X and W are invertible in property (3).
According to (3), (4), and the Kronecker product properties, the Riccati dynamical equation (2) can be rewritten as follows:(5)vecP˙=AvecP+vecU+vecUT+vecGx+vecGxT,vecY=C1vecP,where A=Θ⊗IN+IN⊗Θ, C1=ϒ⊗IN, and IN denotes the identity matrix of order N.Assumption 1.
For the connection relationships subsystem (2), we assume that the double matrix Θ,ϒ is completely stable.
If Assumption1 holds, then we can obtain a matrix K∈RN×N1, which can make Θ+Kϒ to be a Hurwitz stable matrix. Thus, as long as any matrix Q>0 is given, there must be one and only one positive definite matrix M∈RN×N that satisfies the following Lyapunov equation:(6)Θ+KϒTM+MΘ+Kϒ=−Q.Lemma 1.
If Assumption1 is true, then the following Lyapunov equations are obtained:(7a)Θ⊗ΙN+K1C1TM˜+M˜Θ⊗ΙN+K1C1=−Q1,(7b)IN⊗Θ+K2C2TM˜+M˜ΙN⊗Θ+K2C2=−Q2,where M˜=M⊗M, Q1=M⊗Q, Q2=Q⊗M, K1=K⊗IN, K2=IN⊗K, and C2=IN⊗ϒ. Clearly, M˜>0, Q1>0, and Q2>0.Proof.
If Assumption1 is true, then we can obtain (6), by which the following equations can be obtained:(8a)Θ+KϒTM⊗IN+MΘ+Kϒ⊗IN=−Q⊗IN,(8b)IN⊗Θ+KϒTM+IN⊗MΘ+Kϒ=−IN⊗Q.
Using the properties of Kronecker product, (8a) and (8b) can be rewritten as(9a)Θ⊗IN+K⊗INϒ⊗INTM⊗IN+M⊗INΘ⊗IN+K⊗INϒ⊗IN=−Q⊗IN,(9b)IN⊗Θ+IN⊗KIN⊗ϒTIN⊗M+IN⊗MIN⊗Θ+IN⊗KIN⊗ϒ=−IN⊗Q.
Thus, we can obtain(10a)Θ⊗IN+K1C1TM⊗IN+M⊗INΘ⊗IN+K1C1=−Q⊗IN,(10b)IN⊗Θ+K2C2TIN⊗M+IN⊗MIN⊗Θ+K2C2=−IN⊗Q.
If both sides of equalities (10a) and (10b) are multiplied by IN⊗M and M⊗IN from right, respectively, then we can obtain(11a)Θ⊗IN+K1C1TM⊗ININ⊗M+M⊗INΘ⊗IN+K1C1IN⊗M=−Q⊗ININ⊗M,(11b)IN⊗Θ+K2C2TIN⊗MM⊗IN+IN⊗MIN⊗Θ+K2C2M⊗IN=−IN⊗QM⊗IN.
It is noticed thatIN⊗MM⊗IN=M⊗M=M⊗ININ⊗M. Therefore, equalities (11a) and (11b) can be rewritten as follows:(12a)Θ⊗IN+K1C1TM⊗M+M⊗MΘ⊗IN+K1C1=−Q⊗M,(12b)IN⊗Θ+K2C2TM⊗M+M⊗MIN⊗Θ+K2C2=−M⊗Q.
This completes the proof of Lemma1.Assumption 2.
The coupling matrixGx in subsystem (2) satisfies Gx=M−1ϒTΨxM−1, where Ψx∈RN1×N and Ψx≤ρtx2,ρt>0.


## 3. The State Observer Design

In this paper, the stateP in subsystem (2) cannot be measured accurately, which implies that the states of structural balance controller U cannot contain P directly. Therefore, it is necessary to design an observer to estimate the state P in (2), and the definition of the state observer about subsystem (2) is given as follows.Definition 3.
For a given matrix dynamical systemP^˙=FP^,Y,x,ρ, if the state of node subsystem (1) can be measured accurately and P−P^⟶t⟶+∞0 holds, then the matrix dynamical system P^˙=FP^,Y,x,ρ is called a state observer of subsystem (2).
If Assumptions1 and 2 are true, we can design the state observer of subsystem (2) as follows:(13)P^˙=Θ+KϒP^+P^Θ+KϒT+U+UT+LP^,Y,x,ρ−KY−YTKT.
Usingvec⋅ operator for (13), the following equation can be obtained:(14)vecP^˙=A+K1C1+K2C2vecP^+vecU+vecUT+vecLP^,Y,x,ρ−K1vecY−K2vecYT,where P^ represents the estimated value of the state P in subsystem (2), and the robust term LP^,Y,x,ρ=Ω+ΩT,ϒP^≠Y0,ϒP^=Y, where Ω=ρtx2M−1ϒTY−ϒP^M−1/Y−ϒP^. Clearly, vecLP^,Y,x,ρ=vecΩ+vecΩT,C1vecP^≠vecY0,C1vecP^=vecY. At the same time, we note that the equality Y−ϒP^=YT−P^ϒT=vecY−ϒP^=vecYT−P^ϒT holds, then vecΩ=ρtx2M˜−1C1TvecY−C1vecP^/vecY−C1vecP^ and vecΩT=ρtx2M˜−1C2TvecYT−C2vecP^/vecYT−C2vecP^ can be obtained.
In this paper, the estimation error is denoted byE=P−P^. By using properties of vec⋅ operator and Kronecker product, the following error system can be obtained:(15)vecE˙=AvecP−A+K1C1+K2C2vecP^+vecLP^,Y,x,ρ−K1vecY−K2vecYT=A+K1C1+K2C2vecE+M˜−1C1TvecΨx+M˜−1C2TvecΨxT−vecLP^,Y,x,ρ.Lemma 2.
If Assumptions 1 and 2 are true, then the matrix dynamical system (13) is the state observer of subsystem (2).Proof.
Consider the following positive definite function:(16)V1=12vecETM˜vecE.
Then, the orbit derivative ofV along (16) reads as(17)V˙1=vecETM˜vecE˙=vecETM˜A+K1C1+K2C2vecE+M˜−1C1TvecΨx+M˜−1C2TvecΨxT−vecLP^,Y,x=vecETM˜Θ⊗ΙN+K1C1vecE+vecETM˜ΙN⊗Θ+K2C2vecE+vecETM˜M˜−1C1TvecΨx+M˜−1C2TvecΨxT−vecLP^,Y,x=12vecETΘ⊗ΙN+K1C1TM˜+M˜Θ⊗ΙN+K1C1vecE+12vecETΙN⊗Θ+K2C2TM˜+M˜ΙN⊗Θ+K2C2vecE+vecETM˜M˜−1C1TvecΨx+M˜−1C2TvecΨxT−vecLP^,Y,x=−12vecETQ1vecE−12vecETQ2vecE+vecETC1TvecΨx+vecETC2TvecΨxT−vecETM˜vecΩ+vecΩT,C1vecP^≠vecY0,C1vecP^=vecY≤−12vecETQ1vecE−12vecETQ2vecE+vecETC1TvecΨx+vecETC2TvecΨxT−vecETM˜vecΩ+vecETM˜vecΩT,C1vecP^≠vecY0,C1vecP^=vecY≤−12vecETQ1vecE−12vecETQ2vecE+ρtx2vecETC1T+ρtx2vecETC2T−ρtx2C1vecE+C2vecE,C1vecP^≠vecY0,C1vecP^=vecY=−12vecETQ1vecE−12vecETQ2vecE≤0.
From inequality (17), we can obtain that the estimation error matrix E is bounded and E⟶t⟶+∞0. Thus, Theorem 1 is proved.


## 4. The Structural Balance Controller Design

Although we cannot directly use stateP in subsystem (2) to design the structural balance controller U, but the state of subsystem (1), the state estimation information P^ in state observer (13) and the output Y in subsystem (2) can be used in the structural balance controller U.Firstly, some useful definitions about structural balance are given as follows.Definition 4.
A symmetric matrixS=sαβ∈Rm×m is structurally balanced if the inequality sαβsβρsρα>0 holds, where α,β,ρ=1,2,…,m and m≥3.
According to Definition4, the asymptotic structural balance of complex dynamical networks can be defined as follows.Definition 5.
Consider the complex dynamical network, which is composed of subsystems (1) and (2). For a given structural balance matrix P∗, if subsystem (2) can asymptotically track the matrix P∗ under the influences of controller U, i.e., Pt⟶t⟶+∞P∗, then this complex dynamical network is called as structural balance.
Therefore, the control objective in this paper is given as follows.
Control objective: for the complex dynamical network, which consists of subsystems (1) and (2), we need try to design the controller UP^,Y,x,ρ in subsystem (2), such that the phenomenon of structural balance emerges in this complex dynamical network.
In order to achieve the above control objective, the following control scheme is proposed for subsystem (2):(18)U=−Θ+ΚϒP∗+ΚY+Λ,where Λ=−ρtxM˜−1ϒTY−ϒP^M˜−1/vecY−C1vecP^,Y≠ϒP^0,Y=ϒP^, it is easy to see that LP^,Y,x,ρ=−Λ−ΛT.
According to (3), (4) and the properties of vec⋅ operator and Kronecker product, equation (18) can be rewritten as(19)vecU=−Θ⊗IN+K1C1vecP∗+K1vecY+vecΛ,(20)vecUT=−IN⊗Θ+K2C2vecP∗+K2vecYT+vecΛT.
LetP¯=P^−P∗, we can deduce from (14) that(21)vecP¯˙=A+K1C1+K2C2vecP¯+P∗+vecU+vecUT+vecLP^,Y,x,ρ−K1vecY−K2vecYT.Lemma 3.
The estimation stateP^ in observer (13) can asymptotically track the structural balance matrix P∗ by using controller (18).Proof.
The following equation can be obtained from Lemma1:(22)A+K1C1+K2C2TM˜+M˜A+K1C1+K2C2=−Q1+Q2.
Then, the following positive definite function is considered for error system (21):(23)V2=12vecP¯TM˜vecP¯.
Calculating the time derivative ofV2 along (21) gives that(24)V˙2=vecP¯TM˜vecP¯˙=vecP¯TM˜A+K1C1+K2C2vecP¯+P∗+vecU+vecUT+vecLP^,Y,x,ρ−K1vecY−K2vecYT=vecP¯TM˜A+K1C1+K2C2vecP¯+vecP¯TM˜A+K1C1+K2C2vecP∗−Θ⊗IN+K1C1vecP∗+K1vecY+vecΛ−IN⊗Θ+K2C2vecP∗+K2vecYT+vecΛT+vecLP^,Y,x,ρ−K1vecY−K2vecYT=vecP¯TM˜A+K1C1+K2C2vecP¯=12vecP¯TM˜A+K1C1+K2C2+A+K1C1+K2C2TM˜vecP¯=−12vecP¯TQ1+Q2vecP¯<0.
The inequality (24) shows that vecP¯⟶t⟶∞0. Thus, Lemma 3 is completely proved.Theorem 1.
For the complex dynamical network composed of subsystems (1) and (2), if Assumptions 1 and 2 are true, then the state P in subsystem (2) can asymptotically track a given structural balance matrix P∗ by the effect of controller (18), such that the structural balance of this network is achieved.Proof.
From Lemmas2 and 3, we can obtain that limt⟶∞Pt−P^t=0 and limt⟶∞P^t−P∗=0, respectively. Therefore, it is not difficult to get that limt⟶∞Pt−P^t+limt⟶∞P^t−P∗=limt⟶∞Pt−P∗=0, this result proves Theorem 1 is true.


## 5. Simulation Example

In this paper, a continuous analog Hopfield network [36] is considered, which is composed of 10 neurons (nodes), and the ith neuron can be described as(25)x˙i=−1τixi+1CiIi+1Ci∑j=1NpijHjxj,where xi∈R is the state of the ith neuron; Ci expresses the input capacitance of the ith neuron; the decay term τi=RiCi is a positive constant, in which Ri is the effective resistance of the ith neuron; Ii is the input current from other neurons external to the circuit; pij=pji represents the strength of synapse between the ith and the jth neuron; and the coupling function Hjxj=1−e−xj/1+e−xj. Thus, network (25) can be described by the nodes subsystem (1) if we choose fixi=−1/τixi+1/CiIi and ci=1/Ci.Then, we assume that the changes of the strengthpij satisfy the Riccati differential equation (2), and choose Ψx=x1Tx1x1Tx2⋯x1Tx10x2Tx1x2Tx2⋯x2Tx10⋮⋮⋱⋮xN1Tx1xN1Tx2⋯xN1Tx10, i,j=1,2,…,10. If we choose N1=5, then the matrices Θ∈R10×10 and ϒ∈R5×10 in (2) can be generated by the following rules in Matlab:(i)
Each element in matricesΘ∗∈R10×10 and ϒ∗∈R5×10 is a randomly generated integer in range −1,1.(ii)
LetΘ=a⋅Θ∗ and ϒ=a⋅ϒ∗, where a is a known positive constant.(iii)
If the matricesΘ and ϒ generated in Step (ii) satisfy Assumption 1, proceed to the next step or repeat Step (i).(iv)
Each element in column vectorz∈R10 is +1 or −1, which is randomly generated. Then, we can choose P∗=b⋅zzT, where b>0. In this paper, we choose a=0.025,b=5 in the process of simulation.Therefore, we can obtain the matricesK and M˜ by solving the Lyapunov equation (6). Moreover, the node subsystem (1) is determined by the dynamical equation (25), and we can choose the following simulation parameters: Ri=1, Ci=0.1i, Ii=5cost, ρt=0.2, and Q=100I10. Finally, initial values of states xi0 and pij0=pji0, i,j=1,2,…,10, are given by randomly generating numbers in the range −3,3. The simulation results are shown in Figures 1–5 .Figure 1
The state response curves of the nodes subsystem.Figure 2
The response curves of the connection matrixPt.Figure 3
The response curves of the estimation matrixP^t.Figure 4
The response curves of the estimation error matrixE.Figure 5
The response curves of the tracking error matrixP−P∗.From Figures1–5, we can obtain the following conclusions:(i)
From Figures2–4, we can see that the estimation error converges asymptotically to zero. According to Definition 3, we know that the Riccati dynamical equation (13) is a state observer of subsystem (2), and the state observer is effective.(ii)
Figures2 and 5 show that subsystem (2) can asymptotically track the structural balance matrix P∗, which can lead to the phenomenon of structural balance in the network. Compared with other methods of structural balance control [14, 33, 34], the biggest difference is that we can directly design the structural balance controller for the connection relationship subsystem based on the estimation information of the state observer. Therefore, the controller proposed in this paper is simpler, and the requirements of form about the coupling Gx in subsystem (2) are lower.(iii)
From Figures1 and 2, it is easy to see that the neuron (nodes) subsystem (25) and the connection relationship subsystem (2) are mutually coupled. It is worth noting that the states of the nodes are used in state observer (13) and controller (18), and it implies that the dynamic changes of the nodes can affect the observer’s estimated error and the structural balance of the network. On the contrary, the dynamic changes of the connection relationships will also cause the states of nodes to be changed via the effective coupling.


## 6. Conclusion

In this paper, we have proposed a complex dynamical network model, which is composed of the node subsystem and the connection relationship subsystem, and the two subsystems are coupled with each other. Different from the existing research achievements on the state estimation of the node subsystem, we mainly focus on the state estimation of the connection relationship subsystem with outputs and design a state observer to estimate the state of the connection relationships subsystem in this paper. This encouraged us to directly design a structural balance controller for the connection relationship subsystem by using the state estimation information of the connection relationships. Based on this, the structural balance controller proposed in this paper is simpler than the existing methods of structural balance control. Therefore, the design methods of state estimation and structural balance control proposed in this paper can enrich the research results about the state estimation and structural balance of complex dynamical networks.



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