Generalized Mutual Synchronization between Two Controlled Interdependent Networks

and Applied Analysis 3 Assumption 2. Suppose that the vector function g(⋅) is Lipschitz continuous, namely, for any x ∈ R, y ∈ Rn and a constant μ > 0, the following inequality holds: 󵄩󵄩󵄩󵄩g (y) − g (x) 󵄩󵄩󵄩󵄩 ≤ μ 󵄩󵄩󵄩󵄩y − x 󵄩󵄩󵄩󵄩 . (5) Assumption 3. Suppose that the time-varying delays τ 1 (t), τ 2 (t) are continuous differentiable functions with 0 ≤ τ 1 (t), τ 2 (t) ≤ h < ∞ and 0 ≤ ̇ τ 1 (t) ≤ ε 1 < 1. Clearly, this assumption holds for constant τ 1 (t), τ 2 (t). Remark 4. Assumptions 2 and 3 are both general assumptions, which hold for a broad class of real-world chaotic systems, such as Lorenz system, Chua’s oscillator, Chen system, and Lü system [28]. Hence, in the following sections, we always assume that both assumptions hold. Lemma 5 (see [26]). If there are any vectors x, y ∈ R, then the following inequality is true: xTy ≤ 1 2 xTx + 1 2 yTy. (6) 3. Generalized Mutual Synchronization Criteria In this section, by designing appropriate adaptive controllers, we can establish some sufficient conditions to insure the generalized mutual synchronization of the proposed general model in Section 2. Obviously, we can deduce some similar criteria for any simple or typical examples from this general model. Combining (1) and (2) and (3), we can express error system of controlled interdependent networks A and B in terms of ė i (t) = ẏ i (t) − Jẋ i (t) = g (y i (t)) − Jf (x i (t)) + b i N


Introduction
In recent years, extensive efforts have been devoted to understanding the properties of complex networks [1][2][3][4][5].Particularly, as one of the most interesting and significant collective behaviors in real world, synchronization in complex dynamical networks has received increasing interest owing to its many potential applications in nature, socioeconomic systems, or engineering [6].In the existing literature, it has been recognized that the network topology plays a significant role in synchronizability of diffusively coupled complex networks [7,8].Also, by using some effective control schemes, a variety of synchronization phenomena have been discovered in various complex networks (see [9][10][11][12][13][14][15][16][17][18] and relevant references therein).However, the studies mentioned above focused almost exclusively on the inner synchronization inside a single, noninteracting network.
Li et al. [19] studied the outer synchronization (in this paper, we call it mutual synchronization to be defined in Section 2) referring to the synchronization between two or more networks.However, to the best of our knowledge, it can be realized mainly by the open-plus-closed-loop method [19,20] or based on the drive-response concept [21][22][23][24][25][26][27] considering only the intranetwork coupling of network itself.Zheng et al. [28] and Wu et al. [29] further studied the outer synchronization between two complex networks considering two kinds of internetwork coupling, but nevertheless, they both still derived the synchronization criteria based on driveresponse concept and did not place the outer synchronization in the context of interdependent networks.
It is well known that many real-world network systems do interact with and depend on each other; for instance, various infrastructures such as transportation, water supply, fuel, and power stations are coupled together; realistic neuronal networks have a clustered structure and they can be viewed as interdependent networks; the epidemic can spread between the coupled networks of the infection layer and the prevention layer; dealing with secure information and cryptography, one can couple two systems to achieve the mutual synchronization, and so forth.Recently, Buldyrev et al. [30] studied the interdependent networks by presenting future smart grid as a real-life example, where the electrical power grid depends on the information network for control and the information network depends on the electrical power grid for their electricity supply.Then, Mei et al. [31] emphasized that it was urgent to research interdependent networks theory for smart grid.Also, Brummitt et al. [32] demonstrated how interdependence affected cascades of load using a multiple branching process approximation.In a word, efforts have been directed to the cascading failures and robustness of interdependent networks [33][34][35][36][37].In general, it has been recognized that interdependent topologies, especially interlinking strategy and internetwork coupling strength, play a vital role in cascading behaviors and robustness of interdependent networks.Analogously, this motivates us to attempt to explore the effects of interdependent topologies on the mutual synchronization between two interdependent networks.
Quite recently, Um et al. [38] placed synchronization behavior in the context of interdependent networks, where the one-dimensional regular network is mutually coupled to the WS small-world network.Based on the mean-field analytic approach, it has been revealed that the internetwork coupling and the intranetwork coupling play different roles in the synchronizability of the WS network.However, it is still limited to inner synchronization in one of the two interdependent networks and hence it is necessary and significant to study the mutual synchronization between two controlled interdependent networks.
The major contributions of our work are as follows.First, we propose the general model of two controlled interdependent networks  and , which take into account not only the intranetwork coupling, but also the time-varying internetwork delays coupling.Second, we place the synchronization in the context of two controlled interdependent networks and study the generalized mutual synchronization of the proposed model.Third, in the numerical examples, to explore the potential application in smart grid, we couple the NW small-world network described by chaotic power system nodes and the scale-free network described by Lorenz chaotic systems following two interdependent interlinking strategies, respectively.Finally, we verify the influences of intranetwork and internetwork coupling and internetwork delays on the controlled mutual synchronizability, which can help to design the optimal interdependent networks.
The remaining part of this paper is organized as follows.Section 2 introduces some useful mathematical preliminaries and proposes the general model of two controlled interdependent networks.The generalized mutual synchronization is investigated and the main theoretical results of this paper are given in Section 3. In Section 4, two numerical examples are provided to explore the potential application in smart grid and to illustrate the correctness and effectiveness of the theoretical results.Finally, some conclusions and further work are given in Section 5.

Preliminaries and Model Presentation
2.1.Notations.The standard mathematical notations will be utilized throughout this paper.Let R ∈ (−∞, +∞), R  be the -dimensional Euclidean space and let R × be the space of  ×  real matrices; I  ∈ R × denotes the -dimensional identity matrix; we use A  or x  to denote the transpose of the matrix A or the vector x, respectively;  max is the maximum eigenvalue of corresponding real symmetric matrix; ‖x‖ = √ x  x stand for the 2-norm of the vector x; ⨂ presents the Kronecker product of two matrices.

Model of Two Controlled Interdependent Networks.
For simplicity and without loss generality, we consider the following model of two controlled interdependent networks (1) and (2) (we call networks  and , respectively, in this paper) consisting of  identical nodes with time-varying internetwork delays coupling.The dynamical equations for the model of controlled interdependent networks  and  can be given by where x  () = ( 1 (),  2 (), . . .  ())  ∈ R  (y  () = ( 1 (),  2 (), . . .  ())  ∈ R  ) is the state variable of the th node in network () at time ;  : R + × R  → R  ( : R + × R  → R  ) is a smooth vector function; A = (  ) × (B = (  ) × ) stands for the intranetwork coupling matrix describing the topological structure of the network (); namely, if there is a connection from node  to node  in network (), then   (  ) = 1; otherwise,   (  ) = 0; however, C = (  ) × (or D = (  ) × ) is the internetwork coupling matrix representing the direct interaction from  in network  to  in network  (or from  in network  to  in network ); that is, if there exists a connection from  in network  to  in network  (or from  in network  to  in network ), then   (  ) = 1; otherwise,   (  ) = 0;   (  ) and   (  ) are the intranetwork and internetwork coupling strength for node , respectively; ) is an inner coupling matrix describing the interactions between the coupled variables;  1 (),  2 () are the time-varying internetwork coupling delays between networks  and , respectively; u  () ∈   are the nonlinear controllers to be designed later for the mutual synchronization.
Remark 4. Assumptions 2 and 3 are both general assumptions, which hold for a broad class of real-world chaotic systems, such as Lorenz system, Chua's oscillator, Chen system, and Lü system [28].Hence, in the following sections, we always assume that both assumptions hold.
Lemma 5 (see [26]).If there are any vectors x, y ∈ R  , then the following inequality is true:

Generalized Mutual Synchronization Criteria
In this section, by designing appropriate adaptive controllers, we can establish some sufficient conditions to insure the generalized mutual synchronization of the proposed general model in Section 2. Obviously, we can deduce some similar criteria for any simple or typical examples from this general model.
Combining (1) and ( 2) and (3), we can express error system of controlled interdependent networks  and  in terms of where J =   (x  ) is the Jacobian matrix of the function   (x  ).Remark 6.From (7), one can find that adding appropriate controller to nodes is an alternative method to obtain mutual synchronization between two networks.In this paper, we thus mainly focus on the controlled mutual synchronization between two networks in the general context of two interdependent networks.Therefore, the intranetwork coupling matrices A and B and the internetwork coupling matrices C and D can be chosen arbitrarily, meaning that it is not necessary for assuming diffusivity, symmetry, or irreducibility of the matrices A, B, C, and D. In addition, the topology structure, node dynamics, and dimension of state vector of one network can be different from the other.
Remark 7. It is well known that the time delays commonly exist in node dynamics, intranetwork coupling, and internetwork coupling.However, we just consider the time-varying internetwork coupling delays regardless of the others to explore the effects of internetwork coupling behavior on the mutual synchronization.It is noted that many networks of interest, like the Kuramoto model, have nonlinear coupling functions.Similarly, for simplicity, we just consider the linear intranetwork and internetwork coupling.Theorem 8. Suppose that Assumptions 2 and 3 hold and that the adaptive controllers (8) and the corresponding update laws (9) are added to the error system (7).Thus, generalized mutual synchronization between controlled interdependent networks  and  with time-varying internetwork delays coupling can be asymptotically realized.Consider where   are the time-varying feedback gain and   are arbitrary positive constants.

Remark 9.
From the proof of the Theorem 8, we know that () is positive definite, V() is negative definite, and lim  → ∞ e  () = 0.According to Lyapunov stability theory, we can also get that the synchronization state e  () = 0 is asymptotically stable.
Remark 10.It is noted that ( 17) is just a sufficient condition, but not the necessary one for the mutual synchronization between controlled interdependent networks  and .
Based on Theorem 8, we can further obtain some similar synchronization criteria in the following two corollaries.
Combining ( 23) and ( 24), we find that the values of   () are irrelevant to  2 (),   , and   under the action of the proposed adaptive controllers ( 8) and ( 9).Thus, in the following sections, it is reasonable not to consider the effects of  2 (),   , and   on the mutual synchronization between controlled interdependent networks  and .

Numerical Simulations and Results
In this section, two numerical examples and their simulations are given to illustrate the correctness and effectiveness of the theoretical results obtained in the previous sections and to identify the factors that influence the mutual synchronizability.
To measure the speed and performance of mutual synchronization process, we define Actually, is the 2-norm of the synchronization error e(), 0 <  < +∞.Thus, the values of ‖e()‖ in the initial stage and at the end of simulations imply the mutual synchronization speed and performance, respectively.It should be particularly noted that, in all of the following simulations, the main figures and insets describe the values of ‖e()‖ during 0 ≤  < 5 and at the end of simulations ( = 5), respectively.
For simplicity and for comparing, we further assume that the internetwork coupling links are bidirectional and the coupling strength of each node is equal; that is,   = ,   = ,   = , and   = .From Remark 14, we know that the time evolutions of e  () are not relevant to  2 (),   , and   ; thus, it is also reasonable to assume  =  = ,  1 () =  2 () = () to simulate the influences of internetwork coupling strength and delays on the mutual synchronizability.Here, we employ the following two interlinking strategies to produce the interdependency matrices C and D in the two examples respectively.
(i) One-to-one support dependence interlinking strategy [30] (strategy I for short): node   in network  only depends on node   in network  and vice versa.
(ii) Multiple support dependence interlinking strategy [37] (strategy II for short): node in network  may randomly depend on more than one node in network  and vice versa.
Example 15.In this example, we generate the interdependency matrices C and D following the strategy I and design the adaptive controllers according to Theorem 8.When  =  =  = 1, () = 0.5, the mutual synchronization errors e  () are depicted in Figure 1, which shows that controlled interdependent networks  and  can easily achieve the generalized mutual synchronization using the designed controllers.Next, we further simulate the influences of internetwork delays and intranetwork and internetwork coupling strength on the mutual synchronizability between the networks  and .We  Example 16.In this example, we produce the interdependency matrices C and D following the strategy II.To measure the effect of the number of interlinking edges on the mutual synchronizability, we define ⟨⟩ as the average number of interlinking edges for each node in network  and the same to network .We conduct similar simulations as those in Example 15.First, we set  =  =  = 1, ⟨⟩ = 3, () =   /(1 +   ); thus, the time evolutions of the synchronization errors e  () are depicted in Figure 6, which shows that interdependent networks  and  can achieve the generalized mutual synchronization successfully.Then, Figures 7, 8 ).In addition, Figure 11 implies that, to some extent, increase of ⟨⟩ is equivalent to the increase of internetwork coupling strength .

Conclusions and Future Work
In this paper, we extend previous research on the outer synchronization between two complex networks to our work on generalized mutual synchronization between two controlled interdependent networks by considering the time-varying internetwork delays coupling.Our model and relevant results are general and can be easily extended to other interdependent networks because there are not any constraints imposed on the intranetwork and internetwork coupling configuration matrices.Based on Lyapunov theory and corresponding mathematical techniques, some sufficient criteria have been derived to guarantee that the proposed interdependent networks model is asymmetrically synchronized.Two numerical examples have been provided to illustrate the feasibility and effectiveness of the theoretical results and to further simulate the effects of internetwork delays, intranetwork and internetwork coupling strength on the mutual controlled synchronizability.In comparison, we find that, under the proposed adaptive controllers, the intranetwork coupling strength enhances the mutual synchronization, while the internetwork coupling delays and coupling strength suppress it.This indicates that the synchronization phenomenon in interdependent networks is different from that in a single network, which highlights the necessity and significance of considering the mutual synchronization in the context of interdependent networks.Thus, with the help of our findings, one can further understand the mutual synchronization phenomenon in two interdependent networks and design interdependent networks with optimal mutual synchronizability for many potential practical applications.However, the mutual synchronization between two interdependent networks is extremely complex, and we cannot consider all the factors that influence the synchronizability altogether.Also, our theoretical and numerical results are still conservative and the proposed control schemes are still a bit complicated because of the generality of the model.Therefore, how to simplify the control laws and reduce the number of controlled nodes is another important topic and remains to be researched in future.Thus, utilizing the designed controller, one can derive the synchronization conditions based on Lyapunov function approach, which is widely used in dynamic system analysis and design by some recent articles [40][41][42][43][44].

Figure 2 :
Figure 2: The curves of ‖e()‖ for the networks  and  interlinked following strategy I with  =  =  = 1 and different internetwork delays ().