Research article

A new structural entropy measurement of networks based on the nonextensive statistical mechanics and hub repulsion


  • The structure properties of complex networks are an open issue. As the most important parameter to describe the structural properties of the complex network, the structure entropy has attracted much attention. Recently, the researchers note that hub repulsion plays an role in structural entropy. In this paper, the repulsion between nodes in complex networks is simulated when calculating the structure entropy of the complex network. Coulomb's law is used to quantitatively express the repulsive force between two nodes of the complex network, and a new structural entropy based on the Tsallis nonextensive statistical mechanics is proposed. The new structure entropy synthesizes the influence of repulsive force and betweenness. We study several construction networks and some real complex networks, the results show that the proposed structure entropy can describe the structural properties of complex networks more reasonably. In particular, the new structural entropy has better discrimination in describing the complexity of the irregular network. Because in the irregular network, the difference of the new structure entropy is larger than that of degree structure entropy, betweenness structure entropy and Zhang's structure entropy. It shows that the new method has better discrimination for irregular networks, and experiments on Graph, Centrality literature, US Aire lines and Yeast networks confirm this conclusion.

    Citation: Fu Tan, Bing Wang, Daijun Wei. A new structural entropy measurement of networks based on the nonextensive statistical mechanics and hub repulsion[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 9253-9263. doi: 10.3934/mbe.2021455

    Related Papers:

    [1] Xin-You Meng, Yu-Qian Wu . Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696. doi: 10.3934/mbe.2019133
    [2] Hui Wang . Prescribed-time control of stochastic high-order nonlinear systems. Mathematical Biosciences and Engineering, 2022, 19(11): 11399-11408. doi: 10.3934/mbe.2022531
    [3] Jian Meng, Bin Zhang, Tengda Wei, Xinyi He, Xiaodi Li . Robust finite-time stability of nonlinear systems involving hybrid impulses with application to sliding-mode control. Mathematical Biosciences and Engineering, 2023, 20(2): 4198-4218. doi: 10.3934/mbe.2023196
    [4] Jiabao Gu, Hui Wang, Wuquan Li . Output-Feedback stabilization for stochastic nonlinear systems with Markovian switching and time-varying powers. Mathematical Biosciences and Engineering, 2022, 19(11): 11071-11085. doi: 10.3934/mbe.2022516
    [5] Guodong Zhao, Haitao Li, Ting Hou . Survey of semi-tensor product method in robustness analysis on finite systems. Mathematical Biosciences and Engineering, 2023, 20(6): 11464-11481. doi: 10.3934/mbe.2023508
    [6] Zulqurnain Sabir, Hafiz Abdul Wahab, Juan L.G. Guirao . A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models. Mathematical Biosciences and Engineering, 2022, 19(1): 663-687. doi: 10.3934/mbe.2022030
    [7] Yuanpei Xia, Weisong Zhou, Zhichun Yang . Global analysis and optimal harvesting for a hybrid stochastic phytoplankton-zooplankton-fish model with distributed delays. Mathematical Biosciences and Engineering, 2020, 17(5): 6149-6180. doi: 10.3934/mbe.2020326
    [8] Xin Gao, Yue Zhang . Bifurcation analysis and optimal control of a delayed single-species fishery economic model. Mathematical Biosciences and Engineering, 2022, 19(8): 8081-8106. doi: 10.3934/mbe.2022378
    [9] Yi Zhang, Yue Song, Song Yang . T-S fuzzy observer-based adaptive tracking control for biological system with stage structure. Mathematical Biosciences and Engineering, 2022, 19(10): 9709-9729. doi: 10.3934/mbe.2022451
    [10] Zhiqiang Wang, Jinzhu Peng, Shuai Ding . A Bio-inspired trajectory planning method for robotic manipulators based on improved bacteria foraging optimization algorithm and tau theory. Mathematical Biosciences and Engineering, 2022, 19(1): 643-662. doi: 10.3934/mbe.2022029
  • The structure properties of complex networks are an open issue. As the most important parameter to describe the structural properties of the complex network, the structure entropy has attracted much attention. Recently, the researchers note that hub repulsion plays an role in structural entropy. In this paper, the repulsion between nodes in complex networks is simulated when calculating the structure entropy of the complex network. Coulomb's law is used to quantitatively express the repulsive force between two nodes of the complex network, and a new structural entropy based on the Tsallis nonextensive statistical mechanics is proposed. The new structure entropy synthesizes the influence of repulsive force and betweenness. We study several construction networks and some real complex networks, the results show that the proposed structure entropy can describe the structural properties of complex networks more reasonably. In particular, the new structural entropy has better discrimination in describing the complexity of the irregular network. Because in the irregular network, the difference of the new structure entropy is larger than that of degree structure entropy, betweenness structure entropy and Zhang's structure entropy. It shows that the new method has better discrimination for irregular networks, and experiments on Graph, Centrality literature, US Aire lines and Yeast networks confirm this conclusion.



    In recent years, the problems of resource shortage and fragile ecological environment have appeared frequently, threatening the survival and development of posterity to a large extent. Therefore, many people have a keen interest in studying biological systems. Considering the maximization of net economic income, a bio-economic model was proposed in [1] which was based on differential-algebraic equation. Biological and economic stability by adding a popular dynamic was studied in [2,3,4]. The problem of how to obtain best harvest in the bio-economic model was studied in [5,6], which provides a theoretical basis for the rational development of biological resources. The optimal cost control problem of Markov jump system was solved in [7,8]. For the purpose of protecting the environment and maintaining the economic development stably and rapidly, it is very urgent and necessary to research the analysis of the bio-economic system.

    Singular systems have a larger application background in bio-economic systems [9]. The singular systems is different from the normal system in that its stability is more complex. It is well-known that a singular system can be stabilized only when it is regular and impulse-free [10,11]. Therefore, much effort was devoted to singular system and its applications. In the recent years, some singular bio-economic system models with stochastic and bifurcation properties are established, which shows that the research of the singular system is very broad and has good development prospects. A singular biological economy markov jump system is proposed in [12], which takes commodity price as markov chain. The bio-economic singular Markov jumping system was studied and the corresponding control design was proposed in [13].

    In natural environment, environmental fluctuation is a very important part of bio-economic system in real life. To a large extent, there are limitations in the application of deterministic methods in mathematical modeling. Therefore, the future dynamics of the system are difficult to accurately predict. In the various dynamic analysis of the system, the stochastic differential equation model is a important part. The ecological population system model was established using randomness in [14]. Some problems of T-S fuzzy system was researched and the applicaton of this type of system in bio-economy was explored in [15,16]. According to the theory of fishery resource economics, a stochastic singular bio-economic system which based on the T-S fuzzy model was established in [17].

    In practice, some systems can maintain asymptotic stability in an infinite time interval but they do not have good transient characteristics. Consequently, it is meaningful to study the transient behavior within a limited time interval. Several sufficient conditions for the continuous-time systems and discrete-time systems to maintain stability in a finite-time are given in [18,19]. In the singular system, the finite-time stability was redefined which have impulsive effects. Then at this time, sufficient conditions were derived in [20]. The conditions which linear stochastic systems can achieve finite-time stability was studied in [21]. The linear matrix inequality theory was used to obtain a series of properties of linear systems, nonlinear systems and stochastic systems in [22,23]. However, up to now, there are few studies on stochastic singular systems with parameter uncertainties and external disturbances. These problems are very important in practical application and also the main content of our research.

    The purpose of this paper is to research stability in finite-time and achieve control of the singular bio-economic system with stochastic fluctuations. It is undoubtedly challenging to control the density of biological population within a certain range and eliminate the influence of some unfavorable factors. This is also the motivation of this paper. The knowledge of the singular stochastic bio-economic system combined with the application of T-S fuzzy control in [24,25]. Firstly, the T-S fuzzy control model which based on the T-S fuzzy control method is established. Then, it provides a new sufficient condition for the system to achieve stochastic stability in finite-time. On this basis, a state feedback controller which can control the biological populations in a limited range through open resource management is designed. Finally, the effectiveness of the method is verified through simulation in the feasible region.

    Notations: The superscript T of a matrix represents its matrix transpose. A is a positive definite matrix if A>0. deg() represents the degree of the determinant. ε(x) represents expectation of stochastic variable x. diag() denotes a diagonal matrix. λmax(A) and λmin(A) stand for the largest eigenvalue and the smallest eigenvalue of matrix A.

    A single kind of dynamic model which has stage structure proposed by [26] is introduced as follows:

    {˙x1(t)=αx2(t)r1x1(t)βx1(t)ηx21(t),˙x2(t)=βx1(t)r2x2(t), (2.1)

    where x1(t) represents the population density of immature species at time t. x2(t) represents the population density of mature species at time t. α represents the inherent growth rate of the immature. β denotes the transition rate which from the immature species grow into mature species. r1 represents the death rate of immature species. r2 represents the death rate of mature species. ηx21(t) reflects restriction on the growth of immature species density.

    The economic profit is often affected by tax, season, market demand, capture costs and other factors. By the bio-economic theory, the sustainable economic profit should be the sustainable total revenue minus the sustainable total cost. If the population captured in the model (2.1) and the economic benefits of the young population captured are considered, and the economic profit m(t) changes with time, then the singular bio-economic system model can be established as follows:

    {˙x1(t)=αx2(t)r1x1(t)βx1(t)ηx21(t),˙x2(t)=βx1(t)r2x2(t),0=E1(p(t)x1(t)c)m(t), (2.2)

    where E1 represtents harvested effort of the immature species, c denotes the cost coefficient so cE1 represents the total cost, p(t) and m(t) represent the price coefficient and the economic benefits of each individual at time t, respectively.

    We notice that there are many random factors in nature and many other random factors caused by human activities, affecting or interfering the changes of immature population density and mature population density in the real environment. It is assumed that the parameters involved in the deterministic model (2.2) are deterministic and have nothing to do with environmental fluctuations. By considering these factors, we can replace the parameters r1 and r2 to introduce randomness into the model. Firstly, it is supposed that the white noise can affects the mortality rate of biological species by r1r1α1ξ(t) and r2r2α2ξ(t). Secondly, it is assumed that the population density will also be directly affected through the external random parameter ω(t).

    Therefore, the stochastic differential-algebraic equations is established as follows:

    {˙x1(t)=αx2(t)r1x1(t)βx1(t)ηx21(t)E1x1(t)+α1x1(t)ξ(t)+x1(t)w(t),˙x2(t)=βx1(t)r2x2(t)+α2x2(t)ξ(t),0=E1(p(t)x1(t)c)m(t), (2.3)

    where α1, α2 are used to represent two different intensities of the white noises. It is assumed that ξ(t) and ω(t) are independent of each other, the mean value is zero and the standard deviation is Gaussian white noises, that is E[ω(t)]=0, E[ω(t)ω(t+τ)]=δ(τ), δ(τ) is the Dirac function. In the context of biological systems, all the coefficients in Eq (2.3) are non-negative.

    There is a positive equilibrium point and a non-positive equilibrium point in system (2.3). In other words, there are two equilibrium points in total, and only the positive balance point is considered in the context of biological. Therefore, this paper only considers the positive equilibrium point under some certain conditions.

    For express more clearly, it is supposed that the equilibrium point is p=(x1x2m). In order to facilitate further study, the following transformations can be used:

    {ς1(t)=x1(t)x1,ς2(t)=x2(t)x2,ς3(t)=m(t)m. (3.1)

    The system (2.3) can be converted to:

    {˙ς1(t)=α(ς2(t)+x2)r1(ς1(t)+x1)β(ς1(t)+x1)η(ς1(t)+x1)2,E1(ς1(t)+x1)+α1(ς1(t)+x1)ξ(t)+(ς1(t)+x1)w(t)˙ς2(t)=β(ς1(t)+x1)r2(ς2(t)+x2)+α2(ς2(t)+x2)ξ(t),0=E1(p(t)(ς1(t)+x1)c)(ς3(t)+m). (3.2)

    The Eq (3.2) is obviously a nonlinear system. Since species density saturation exists, it can be assumed that ςi(t)(i=1,2,3) are bounded. Make the following changes to the system (3.2) to make expression more concise:

    E˙ς(t)=[Ω11α0βr2+α2ξ(t)0E1p(t)01]ς(t)+[αx2r1x1βx1ηx12E1x1+x1α1ξ(t)+x1ω(t)+ς1(t)w(t)βx1r2x2+x2α2ξ(t)E1p(t)x1E1cm], (3.3)

    where

    Ω11=r1βης1(t)2ηx1E1+α1ξ(t)
    ς(t)=[ς1(t)ς2(t)ς3(t)],E=[100010000].

    Let

    z(t)=r1βης1(t)2ηx1E1+α1ξ(t),

    so

    maxz(t)=r1βηςmin1(t)2ηx1E1+α1ξ(t),minz(t)=r1βηςmax1(t)2ηx1E1+α1ξ(t).

    z(t) is expressed as follows by the max-min values

    z(t)=M11(z(t))maxz(t)+M12(z(t))minz(t)

    where M11+M12=1 and M11, M12 denote the membership functions. Given the fuzzy rules as follows:

    Model rule 1:

    If z(t) is M11(z1(t)), then E.ς(t)=A1ς(t)+B1ς(t)w(t)

    Model rule 2:

    If z(t) is M12(z1(t)), then E.ς(t)=A2ς(t)+B2ς(t)w(t)

    The system (3.3) can be changed into Eq (3.8) by using the fuzzy rules:

    Edς(t)=Ahς(t)dt+Bς(t)dw(t) (3.4)

    where Ah=2i=1hi(z(t))Aihi(z(t))02i=1hi(z(t))=1B=B1=B2.

    Definition 3.1. (ⅰ) If there is a constant λ such that det(λEAh)0, then the system (3.4) is regular.

    (ⅱ) If rank(E)=deg(det(λEAh)), then the system (3.4) is impulse-free.

    Definition 3.2. Given a positive definite matrix R, for any two positive numbers l1, l2 then satisfy l1l2, the system (3.4) for (l1,l2,T,R) is stochastic finite-time admissible if

    ε{ςT(0)ETREς(0)}l1ε{ςT(t)ETREς(t)}l2,t[0,T]

    Definition 3.3. [27] Given a stochastic Lyapunov function V(x(t),t), where x(t) satisfies the following equation which is a stochastic differential equation in a stochastic system:

    dx(t)=f(t)dt+g(t)dw(t) (3.5)

    The definition of the weak infinitesimal operator L in the random process is given as {x(t),t>0}:

    LV(x(t),t)=V(x(t),t)t+V(x(t),t)xf(t)+12tr[gT(t)Vxx(x(t),t)g(t)] (3.6)

    Lemma 3.4. ([28] Gronwalls inequality) Given a non-negative function g(t), for any two constants m,n and they satisfy m,n0

    g(t)m+nt0g(s)ds,0tT (3.7)

    then

    g(t)mexp(nt) (3.8)

    Lemma 3.5. (Schur's complement) Exist any real matrix A,B,C, among them BT=B and CT=C>0, the following three conditions are equivalent:

    (i)B+AC1AT<0
    (ii)(BAATC)<0
    (iii)(BAATC)<0

    Lemma 3.6. [29](i) For two orthogonal matrices U and V, if rank(E)=r, then E can be decomposed as follows

    E=U[r00]VT=U[Ir00]vT (3.9)

    where r=diag{δ1,δ2,,δr},δk>0,k=1,2,,r

    Partition U=[U1U2],V=[V1V2],v=[V1rV2] with V2=0,UT2E=0.

    (ii)If P satisfies

    EPT=PET (3.10)

    Then

    ˜P=UTPv=[p11p120p22]. (3.11)

    We can have P110, det(P22)0 if P is nonsingular. Furthermore, P that satisfied Eq (3.10) can be set as follows

    P=EvTYv1+UZVT2 (3.12)

    where Y=diag{P11,Φ},Z=[PT11PT22]T, among this ΦR(nr)×(nr) is an arbitrary parameter matrix.

    (iii) The following equation holds if P is a nonsingular matrix, C and Φ are two positive definite matrices, Y is a diagonal matrix in Eq (3.12), P and E satisfy Eq (3.13)

    P1E=ETC12SC12E (3.13)

    Then the solution of Eq (3.13) can be expressed by S=C12UY1UTC12, where S is a positive definite matrix.

    In this section, we discuss whether the model (3.4) can be stable in a finite-time.

    Theorem 4.1. If there is a non-singular symmetric positive definite matrix Q makes any given time constant T>0 and scalar α>0 satisfy

    PET=EPT0 (4.1)
    P1E=ETR12QR12E (4.2)
    λmax(Q)l1eαTl2λmin(Q)<0 (4.3)

    and satisfy the following matrix inequalities:

    [A1PT+PAT1αEPT(EBPT)TEBPT˜Q]<0 (4.4)
    [A2PT+PAT2αEPT(EBPT)TEBPT˜Q]<0 (4.5)

    where ˜Q=R12Q1R12. Then the system (3.4) is tolerable in a stochastic finite-time for (l1,l2,T,R).

    Proof. Introduce the following Lyapunov function

    V(ς(t),t)=ςT(t)P1Eς(t) (4.6)

    Let L be the infinitesimal generator. We can get the following inequality:

    LV(ς(t),t)<αV(ς(t),t) (4.7)

    In the following, we can prove that the three conditions Eqs (4.7), (4.4) and (4.5) are equivalent. Applying Itˆo formula, we can get

    LV(ς(t),t)=ςT(t)(AThPT+P1Ah+BTP1EB)ς(t) (4.8)

    then

    LV(ς(t),t)αV(ς(t),t)=ςT(t)(AThPT+P1Ah+BTP1EBαP1E)ς(t) (4.9)

    Combining condition Eq (4.2) with ˜Q=R12Q1R12, Eq (4.9) can be transformed as

    LV(ς(t),t)αV(ς(t),t)=ςT(t)(AThPT+P1Ah+BTET˜Q1EBαP1E)ς(t) (4.10)

    Therefore, from Eq (4.10), Eq (4.7) is equivalent to

    AThPT+P1Ah+BTET˜Q1EBαP1E<0 (4.11)

    The left is multiplied by P and on the right multiplied by PT, we get

    PATh+AhPT+PCTET˜Q1EBPTαEPT<0 (4.12)

    It can be obtained that Eq (4.12) is equivalent to Eqs (4.4) and (4.5) by using matrix factorization and Schur's complement lemma. Integrating the left and right sides of Eq (4.7) from 0 to t at the same time and then taking the expected value, we can get

    ε{V(ς(t),t)}<V(ς(0),0)+αt0ε{V(ς(s),s)}ds (4.13)

    From Lemma 3.4, we have

    ε{V(ς(t),t)}<V(ς(0),0)eαt (4.14)

    from inequalities

    ε{V(ς(t),t)}=ε{ςT(t)ETR12QR12Eς(t)}λmin(Q)ε{ςT(t)ETREς(t)} (4.15)

    and

    V(ς(0),0)eαt=ςT(0)ETR12QR12Eς(0)eαtλmax(Q)ςT(0)ETREς(0)eαtλmax(Q)l1e (4.16)

    then, we get

    ε{ςT(t)ETREς(t)}<λmax(Q)λmin(Q)l1eαT (4.17)

    Considering condition Eq (4.3) and inequality (4.17), for the t[0,T], have

    ε{ςT(t)ETREς(t)}<l2 (4.18)

    In practice, the bio-economic system is more or less disturbed by the external environment. For example, the growth of a population can be affected by environmental factors such as the intensity of sunlight and the temperature. This section considers this random environment that affects populations as a zero-mean Gauss white noise. In order to achieve effective planning of capture strategies and maintain the sustainable development of market economy, some measures must be taken to stabilize the biological population. Therefore, control is added to the singular stochastic bioeconomic system model (2.3):

    {˙x1(t)=αx2(t)r1x1(t)βx1(t)ηx21(t)E1x1(t)+α1x1(t)ξ(t)+u(t)+x1(t)w(t),˙x2(t)=βx1(t)r2x2(t)+α2x2(t)ξ(t),0=E1(p(t)x1(t)c)m(t), (4.19)

    where u(t) represents the management degree of open resources and it is a control variable.

    Through the method which is similar to the T-S fuzzy method mentioned in Section 3, we get

    Edς(t)=2i=1hi(z(t))((Aiς(t)+Cu(t))dt+Bς(t)dw(t)) (4.20)

    Consider the following state feedback controller:

    u(t)=2i=1hi(z(t))Giς(t) (4.21)

    By designing the state feedback gain Gi.

    Thus, the corresponding closed-loop system can be expressed as

    Edς(t)=2i=1hi(z(t))2j=1hj(z(t))[(Ai+CGj)ς(t)dt+Bς(t)dw(t)] (4.22)

    where C=[100]T

    Next, design parameters of the fuzzy state feedback controller Eq (4.21), a new sufficient condition for the closed-loop singular stochastic bio-economic system to be stable in a finite-time is given.

    Theorem 4.2. The closed-loop system (4.22) is stochastic finite-time admissible for a state feedback controller Eq (4.21) relative to (l1,l2,T,R), if P is a nonsingular matrix, Q is a symmetric positive definite matrix and any matrix Yj,j=1,2 which can make Eqs (4.1)–(4.3) hold, and the following matrix inequalities can be satisfied:

    [γii(EBPT)TEBPT˜Q]<0,i=j,i,j=1,2 (4.23)
    [γij+γji(EBPT)TEBPT˜Q]<0,i<j,i,j=1,2 (4.24)

    where γij=PATi+YTjCT+AiPT+CYjαEPT and ˜Q=R12Q1R12, Next, we can choose the state feedback Gj=YjPT which we need.

    Proof. First, regularity and impulse-free of the system can be proved. Without loss of generality, denoting

    E=[Ir000]

    where rank(E)=rank(Ir)=rn.

    Suppose that there are two non-singular matrices H and K, then

    HEK=[Ir000]H(Ai+CGj)K=[Ai11Ai12Ai21Ai22]HBK=[B1B200]HTPK1=[P11P12P21P22] (4.25)

    Now, if (E,Ai+CGj) is impulse-free, we can prove the impulselessness of the system (4.22). The system (4.22) is impulse-free if

    rank[EAi+CGj0E]=n+rank(E) (4.26)

    It can be computed from Eq (4.25) that

    rank[EAi+CGj0E]=rank[Ir0Ai11Ai1200Ai21Ai2200Ir00000]=2r+rank(Ai22) (4.27)

    This shows Eq (4.26) is equivalent to n=r+rank(Ai22), then (E,Ai+CGj) is regular and impulse-free if Ai22 is non-singular simultaneously.

    Considering Theorem 4.1 and the system (4.22), the following matrix inequality can be obtained:

    2i=12j=1hi(z(t))hj(z(t))Δij<0 (4.28)

    where

    Δij=P(Ai+CGj)T+(Ai+CGj)PT+PCTET˜Q1EBPTαEPT

    Let

    Yj=GjPT

    Then

    2i=12j=1hi(z(t))hj(z(t))(PATi+YTjC+AiPT+CYj+PCTET˜Q1ECPTαEPT)<0 (4.29)

    Obviously, Eq (4.29) is equivalent to

    2i=1h2i(z(t))(γii+PCTET˜Q1ECPT)+2i=12j=1hi(z(t))hj(z(t))[(γij+γji+PCTET˜Q1ECPT)]<0 (4.30)

    Using matrix decomposition and Lemma 3.5, the above inequality (4.30) is equivalent to inequalities (4.23) and (4.24). Next, by using Eqs (4.2)–(4.3) and similar proof of Theorem 4.1, the system (4.22) is stochastic finite-time admissible for (l1,l2,T,R).

    Remark 1. By designing a fuzzy state feedback controller, the government's management of open resource development can be better expressded. The density of biological population can be controlled within a limited range and some unfavorable phenomena can be eliminated. In real life, managers should take some effective measures, such as adjusting taxes, introducing some preferential policies to stimulate the development of fisheries, reducing environmental pollution and so on. Thus, the population density can be strictly controlled and the economic benefits can be maintained steadily.

    We use the following special circumstances to prove that the results obtained are true and effective. Selection of ecological parameters based on Nile tilapia data from Lake Tanganyika in Africa.

    Consider the finite-time stability of bio-economic model with white noise, we choose the ecological parameters of the appropriate unit:

    α=0.4,r1=0.5,β=0.5,η=0.1α1=0.1,r2=0.1,α2=0.1,p(t)=1E1=0.8,c=3,ξ(t)=1,w(t)=0.1

    Then, the singular bioeconomic system can be obtained:

    {˙x1(t)=0.4x2(t)0.5x1(t)0.5x1(t)0.1x21(t)0.8x1(t)+0.1x1(t)ξ(t)+u(t)+x1(t)w(t),˙x2(t)=0.5x1(t)0.1x2(t)+0.1x2(t)ξ(t),0=0.8(p(t)x1(t)3)m(t), (5.1)

    where

    x1(t)[0,5],x2(t)[0,10],m(t)[0,5]

    We can get the system (5.1) has an equilibrium point p(5,15,1.225) when u(t)=0.

    For achieveing the transition from the equilibrium point to origin, a following fuzzy models can be constructed by using linear transformation (3.1):

    E.ζ(t)=[Ω110.400.50.1+0.1ξ(t)00.8p(t)01]ζ(t)+[5.5+0.5ξ(t)+5w(t)+ζ1(t)w(t)+u(t)11.5ξ(t)4p(t)3.625], (5.2)

    where

    ζ1(t)[5,0],ζ2(t)[15,5],ζ3(t)[1.225,3.775]
    Ω11=2.80.1ζ1(t)+0.1ξ(t)

    we have

    maxz(t)=3.3+0.1ξ(t)minz(t)=2.8+0.1ξ(t)

    By using the fuzzy rules mentioned above, we can execute the fuzzy model as:

    Edς(t)=2i=1hi(z(t))2j=1hj(z(t))[(Ai+CGj)ς(t)dt+Bς(t)dw(t)] (5.3)

    where

    E=[100010000],A1=[3.20.400.5000.801]A2=[2.70.400.5000.801],B=[4.50.50.375]C=[100]T

    Let

    α=0.001,l1=10000,l2=10000000,T=1000,R=I

    Then we get

    G1=[8.657947.25910.0345]G2=[8.622147.27850.0364]

    Therefore, we can get that ε{ζT(t)ETREζ(t)}<10000000, for all t[0,1000].

    From Figure 1, we can see the trajectory of a stochastic singular bio-economic system which is open-loop and considers the white noise. It can be seen that the species density and average price are unstable within a limited time. The economic interests fluctuates randomly in Figure 1, on account of the economic interests is affected by the population unit price and the species density of biological economic model which proposed in this paper in a randomly disturbed environment.

    Figure 1.  Trajectory of the open-loop stochastic singular system.

    From Figure 2, through the state feedback controller Eq (4.21), we can see the trajectory of a stochastic singular bio-economic system which is closed-loop and considers the white noise. It can be seen from Figure 2 that economic profits tend to be stabilize within a limited time.

    Figure 2.  Trajectory of the closed-loop stochastic singular system.

    In this paper, the finite time stability and control of a kind of singular bio-economic system with stochastic fluctuations which based on T-S fuzzy model are studied. Through two theorems, we derived some new sufficient conditions to guarantee the stability of system in finite time. The corresponding controller design method is also given. Finally, the effectiveness of the method is verifies through a numerical simulation. The results are also applicable to other types of systems which is similar to the system of this paper.

    From a biological point of view, the biological species density can be controlled within a certain range through the fuzzy state feedback controller which designed in this paper, eliminating influence of some unfavorable factors, It can better control the density of the population and keep the economic profit stable.

    This work is supported by National Natural Science Foundation of China under Grant 61673099.

    The authors declare there is no conflict of interest.



    [1] G. Gradziska, A. Kulig, J. Jaroslaw, S. Drozds, Complex network analysis of literary and scientific texts, Int. J. Mod. Phys. C, 23 (2012), 1250051. doi: 10.1142/S0129183112500519
    [2] M. Liu, Y. M. Yan, Y. Huang, Complex system and its application in urban transportation network, Sci. Technol. Rev., 25 (2017), 27–33.
    [3] D. J. Watts, S. H. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), 440–442. doi: 10.1038/30918
    [4] D. Wang, J. L. Gao, D. J. Wei, A new belief entropy based on deng entropy, Entropy, 21 (2019), 987–988. doi: 10.3390/e21100987
    [5] T. Wang, L. L. Wu, J. Zhang, Research on correlation properties of urban transit network based on complex network, J. Aca. Mili. Trans., 11 (2009), 10–15.
    [6] L. F. Costa, F. N. Silva, Hierarchical characterization of complex networks, J. Stat. Phys., 125 (2004), 841–872.
    [7] F. N. Silva, L. F. Costa, Local dimension of complex networks, preprint, arXiv: 1209.2476.
    [8] Y. Long, Visibility graph network analysis of gold price time series, Phys. A, 392 (2013), 3374–3384. doi: 10.1016/j.physa.2013.03.063
    [9] D. J. Watts, S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440–440. doi: 10.1038/30918
    [10] A. L. Barabasi, R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512. doi: 10.1126/science.286.5439.509
    [11] C. M. Song, S. Havlin, H. A. Makse, Self-similarity of complex networks, Nature, 433 (1999), 392–395.
    [12] D. J. Wei, B. Wei, Y, Hu, H. X. Zhang, Y. Deng, A new information dimension of complex networks, Phys. Lett. A, 378 (2014), 1091–1094. doi: 10.1016/j.physleta.2014.02.010
    [13] D. J. Wei, X. Y. Deng, X. G. Zhang, Y. Deng, S. Mahadevan, Identifying influential nodes in weighted networks based on evidence theory, Phys. A, 392 (2013), 2564–2575. doi: 10.1016/j.physa.2013.01.054
    [14] C. Christoph, I. Jacopo, A. Alex, B. Ginestra, Centralities of nodes and influences of layers in large multiplex networks, J. Complex. Netw., 6 (2017), 733–752.
    [15] C. Christoph, I. Jacopo, A. Alex, B. Ginestra, Ranking the spreading ability of nodes in network core, Int. J. Mod. Phys. C, 26 (2015), 12305–12310.
    [16] C. Christoph, I. Jacopo, A. Alex, B. Ginestra, Complex networks renormalization: flows and fixed points, Phys. Rev. Lett., 101 (2008), 148701–148704. doi: 10.1103/PhysRevLett.101.148701
    [17] M. A. Serrano, D. Krioukov, M. Boguna, Self-similarity of complex networks and hidden metric spaces, Phys. Rev. Lett., 100 (2008), 078701–078704. doi: 10.1103/PhysRevLett.100.078701
    [18] M. L. Lei, D. J. Wei, A measure of identifying influential community based on the state of critical functionality, Math. Biosci. Eng., 17 (2020), 7167–7191. doi: 10.3934/mbe.2020368
    [19] X. L. Xu, X. F. Hu, X. Y. He, Degree dependence entropy descriptor for complex networks, Adv. Manuf., 1 (2013), 284–287. doi: 10.1007/s40436-013-0034-1
    [20] C. E. Shannon, A mathematical theory of communication, Bell. Syst. Tech. J., 27 (1948), 623–656. doi: 10.1002/j.1538-7305.1948.tb00917.x
    [21] M. L. Lei, L. R. Liu, D. J. Wei, An improved method for measuring the complexity in complex networks based on structure entropy, IEEE Access, 99 (2019), 1–16.
    [22] C. Christoph, I. Jacopo, A. Alex, B. Ginestra, The density-based clustering method for privacy-preserving data mining, Math. Biosci. Eng., 16 (2019), 1718–1728. doi: 10.3934/mbe.2019082
    [23] Y. Z. Yang, L. Yu, X. Wang, S. Y. Chen, Y. Chen, Y. P. Zhou, A novel method to identify influential nodes in complex networks, Int. J. Mod. Phys. C, 31 (2020), 2050021–20500214. doi: 10.1142/S0129183120500217
    [24] K. Anand, G. Bianconi, Entropy measures for complex networks: Toward an information theory of complex topologies, Phys. Rev. E, 80 (2009), 0451021–045104.
    [25] Q. Zhang, M. Z. Li, Y. Deng, A betweenness structure entropy of complex networks, preprint, arXiv: 1407. 0097.
    [26] B. Wang, J. Zhu, D. J. Wei, The self-similarity of complex networks: from the view of degree-degree distance, Mod. Phys. Lett. B, 35 (2021), 21503311–215033113.
    [27] Y. X. Du, C. Gao, Y. Hu, S. Mahadevan, Y. Deng, A new method of identifying influential nodes in complex networks based on topsis, Phys. A, 399 (2014), 57–69. doi: 10.1016/j.physa.2013.12.031
    [28] C. Christoph, I. Jacopo, A. Alex, B. Ginestra, Hierarchical feedback modules and reaction hubs in cell signaling networks, PLoS One, 10 (2015), e0125886. doi: 10.1371/journal.pone.0125886
    [29] J. F. Xu, Y. H. Lan, Centralities of nodes and influences of layers in large multiplex networks, J. Complex. Netw., 6 (2017), 733–752.
    [30] C. M. Song, S. Havlin, H. A. Makse, Origins of fractality in the growth of complex networks, Nat. Phys., 4 (2006), 275–281.
    [31] Q. Zhang, M. Z. Li, Y. Deng, A new structure entropy of complex networks based on nonextensive statistical mechanics, Int. J. Mod. Phys. C, 27 (2016), 1650118. doi: 10.1142/S0129183116501187
    [32] B. Wang, F. Tan, J. Zhu, D. J. Wei, A new structure entropy of complex networks based on nonextensive statistical mechanics and similarity of nodes, Math. Biosci. Eng., 18 (2021), 3718–3732. doi: 10.3934/mbe.2021187
    [33] M. X. Liu, S. S. He, Y. Z. Sun, The impact of media converge on complex networks on disease transmission, Math. Biosci. Eng., 16 (2019), 6335–6349. doi: 10.3934/mbe.2019316
    [34] L. C. Freeman, A set of measures of centrality based on betweenness, Sociometry, 40 (1977), 35–41. doi: 10.2307/3033543
    [35] B. S. Baigrie, Electricity and magnetism: a historical perspective, Greenwood Pub. Group Inc., 2006.
    [36] Y. Xiao, W. Wu, M. Xiong, W. Wang, Symmetry based structure entropy of complex networks, Phys. A, 387 (2007), 2611–2619.
  • This article has been cited by:

    1. Liping Bai, Juan Zhou, Delay-Dependent H
    Control for Singular Time-Varying Delay Systems with Markovian Jumping Parameters, 2022, 41, 0278-081X, 6709, 10.1007/s00034-022-02074-8
    2. Haiying Wan, Xiaoli Luan, Vladimir Stojanovic, Fei Liu, Self-triggered finite-time control for discrete-time Markov jump systems, 2023, 00200255, 10.1016/j.ins.2023.03.070
    3. Jinling Wang, Jinling Liang, Cheng-Tang Zhang, Ang Gao, Dongmei Fan, Event-triggered asynchronous nonfragile guaranteed performance control and l1-gain analysis for state-dependent switched singular positive systems, 2025, 698, 00200255, 121768, 10.1016/j.ins.2024.121768
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2752) PDF downloads(90) Cited by(6)

Figures and Tables

Figures(3)  /  Tables(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog