A New Model for Complex Dynamical Networks Considering Random Data Loss

Model construction is a very fundamental and important issue in the field of complex dynamical networks. With the state-coupling complex dynamical network model proposed, many kinds of complex dynamical network models were introduced by considering various practical situations. In this paper, aiming at the data loss which may take place in the communication between any pair of directly connected nodes in a complex dynamical network, we propose a new discrete-time complex dynamical network model by constructing an auxiliary observer and choosing the observer states to compensate for the lost states in the coupling term. By employing Lyapunov stability theory and stochastic analysis, a sufficient condition is derived to guarantee the compensation values finally equal to the lost values, namely, the influence of data loss is finally eliminated in the proposed model. Moreover, we generalize the modeling method to output-coupling complex dynamical networks. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed model.


Introduction
Complex networks exist in different fields such as Internet, power grids, food web, etc., and have received a great deal attention over the past decades. Researchers have tried to build mathematical models for various types of networks in the real world, some of which focused on the network topology. These complex network models have been studied by graph theory, which are represented by nodes connected by edges. Some classic models, such as the E-R random-graph model [1], the WS small-world model [2], and the BA scale-free model [3], have led research hotspots in complex networks and made outstanding contributions to the development of complex networks.
Subsequently, some researchers realized that they should not only focus on the network's topological connectivity, but should also consider the dynamics of network nodes in order to better understand the dynamical behaviors of various complex networks. Pecora and Carroll [4] constructed a state-coupling model by introducing coupling coefficients and a matrix to link the node connectivity and the node dynamics together. Wang et al. [5] considered a scale-free dynamical network consisting of identical linearly coupled nodes and studied its robustness and fragility of synchronization. Lü et al. [6] introduced a time-varying state-coupling complex dynamical network model, with time-varying coupling configuration matrix and inner-coupling matrix. Differing from the previous studies, Li et al. [7] restricted the inner-coupling matrix as the identity matrix, which means that two coupled nodes are diagonally linked through their corresponding components. These state-coupling complex dynamical network models have been adopted by many follow-up studies.
Many practical complex networks, covering different fields such as communication networks and social networks [8,9], all undertake the objective of information transmission. Especially for

Network Modeling and Preliminaries
The typical discrete-time complex dynamical network is as follows: where i = 1, 2, . . . , N denotes the ith node, x i,k = (x i1,k , x i2,k , . . . , x in,k ) T ∈ R n denotes the state vector of the ith node at time k, A ∈ R n×n is a constant matrix, f (·) : R n → R n is the known nonlinear function, d is the coupling strength, and matrix C = c ij N×N is the coupling configuration matrix.
If there is a link from node j to node i (i = j), then c ij = 1; otherwise, c ij = 0. Assume that matrix C satisfies c ii = − ∑ N j=1,j =i c ij . Γ ∈ R n×n is the inner connecting matrix between two connected nodes. The complex dynamical network model (1) is established in an ideal situation without the consideration of random data loss. Yang et al. [23] considered the data loss taking place in the communication between neighbor nodes, and established the corresponding complex dynamical network model as follows: where b ij,k ∈ R are independent identically distributed Bernoulli random variables. If there is data loss in the link from node j to node i (i = j) at time k, then b ij,k = 0; otherwise, b ij,k = 1. b ii,k = 1 always holds. b ij,k takes 0 or 1 with the probabilities: However, the influence of data loss still exists in the network, and may give rise to adverse effects such as low transmission efficiency or transmission failure. Therefore, we model a complex dynamical network with data loss by introducing an auxiliary observer and compensating for the lost states with the corresponding observer states in the coupling term. The corresponding state-coupling complex dynamical network model is presented as follows: where y i,k ∈ R m are the outputs of the ith node in the network, H i ∈ R m×n denote the output matrices of the ith node,x i,k = (x i1,k ,x i2,k , . . . ,x in,k ) T ∈ R n denote the observation values of x i,k ,ŷ i,k ∈ R m are the outputs of the observer, and K i ∈ R n×m are the observer gains to be determined. Here, we assume the data loss processes on all the interactions between the neighbor nodes are mutually independent. Remark 1. The random data losses bring uncertainty to the network, which can be measured by the entropy in information theory [31,32]. In this paper, the random data losses are described by a set of random variables satisfying the Bernoulli distribution, so the entropy of each Bernoulli random process Since all the Bernoulli random processes are mutually independent, the joint entropy is ∑ N i,j=1 c ij H b ij . (3) is constructed under the assumption that the transmission time between the network and the observer can be neglected. Namely, the transmission time from y i,k to the observer andx j,k to the network is neglectable.

Remark 3.
In order to eliminate the influence of random data loss, we tried to find a kind of compensation method whose compensation value could finally equal to the lost data. Because of the favorable performance on observation, we chose the observer state values as the compensation values and propose the complex dynamical network model (3). Note that the aim of the proposed model (3) is to compensate the lost network states with the observer states after the error convergence is achieved in an unbiased fashion in real-time. The data loss existing before convergence cannot be compensated without bias. We will study the finite-time and the fixed-time asymptotic convergence in future work to speed up convergence.

Remark 4.
In the actual data transmission process, there generally exists a detecting mechanism to judge whether the data are transmitted successfully or not. For example, in the Internet, TCP (Transmission Control Protocol) uses the ACK (acknowledgment character) to acknowledge receipt of a packet. Therefore, we can judge whether b ij,k = 1 or not via a certain detection mechanism and realize the construction of the proposed model.
For the purpose of analyzing the proposed complex dynamical network model (3), an assumption and a lemma are given as follows.

Assumption 1.
There exists a positive constant α such that

Model Analysis
In this section, we analyze the complex dynamical network model (3) proposed in Section 2. An auxiliary observer is introduced in the proposed model (3). Once the data loss happens in a data transmission channel, the corresponding data in the observer will be used to compensate for the lost data. If appropriate observer gains K i are chosen to make the observation statesx i,k approach the network states x i,k , then the observer outputsŷ i,k will approach the network outputs y i,k , the coupling term ∑ N j=1 b ij,k c ij Γx j,k + 1 − b ij,k c ij Γx j,k will approach ∑ N j=1 c ij Γx j,k , and the output feedback term K i (ŷ i,k − y i,k ) will approach the zero matrix. Thus the proposed complex dynamical network model (3) will approach the ideal complex dynamical network model (1), that is, the influence of the random data loss can be eliminated when state observation is achieved.
In the following, we derive a method to determine feasible observer gains K i . Defining e i,k =x i,k − x i,k (i = 1, 2, . . . , N) as the observation errors in the complex dynamical network model (3), one can obtain the following error system: Theorem 1. Suppose that Assumption 1 holds. The error system (5) is asymptotically stable and the error states converge to zero if there exist matrices P i = P T i > 0, S i (i = 1, 2, . . . , N) and a scalar τ > 0 satisfying the following inequality: where Then, the observer gains can be determined by Proof of Theorem 1. Choose the following Lyapunov function: Deriving the difference of V (k), one obtains: and η k = e T kf T k T . Taking the mathematical expectation of ∆V (k), one has: Then, from the Lipschitz condition (Assumption 1), we can get thatf T kf k ≤ α 2 e T k e k , which is equivalent to As the T k is non-positive, (9) is negative definite if and only if there exists a scalar τ > 0 such that E [∆V (k)] < τT k . Hence, the following inequality can be obtained: Using Lemma 1, we can see that (11) is equivalent to (6). According to the Lyapunov stability theory and stochastic analysis, the error system (5) is asymptotically stable (i.e., the error variables will converge to zero), and the observer gains can be obtained by K i = P −1 i S i . The proof is completed.
We can calculate the observer gains K i by solving the LMI (6) and complete the construction of the proposed complex dynamical network model (3). [5][6][7][34][35][36][37][38] have assumed state-coupling (usually diagonal coupling) among the nodes in a network, implying that a node communicates with its connected neighbors by all its state variables. In addition, there exists the output-coupling pattern that each node communicates with neighbors only by its outputs. Here, we generalize the proposed modeling method to a output-coupling complex dynamical network with random data loss:

Remark 5. Many works
where L ∈ R n×m denotes the inner coupling matrix.
The analysis concept of the output-coupling complex dynamical network (12) is similar to that of the proposed model (3), so the detailed analysis is omitted here.

Numerical Simulation
In this section, we give two numerical examples to demonstrate the validity of the proposed discrete-time complex dynamical network model with random data loss. Here, we consider complex dynamical networks generated from the WS small-world network model and the BA scale-free network model due to the universality of the "small-world" and "scale-free" characteristics in most real networks.

Example 1. A state-coupling WS small-world network.
Consider the state-coupling WS small-world network with 10 nodes shown in Figure 1. The corresponding network coupling configuration matrix is: The node dynamic is the following nonlinear system: This satisfies Assumption 1 by α = 0.4. From Figure 2, we know this node dynamic is a non-stationary system.
The initial conditions of x i,k ,x i,k (i = 1, 2, . . . , 10) are respectively taken as the random numbers in the intervals [−2, 2] and [−3, 3]. Then, according to Theorem 1 and using the YALMIP toolbox in Matlab, we can get the P i and the corresponding observer gains K i as follows: Three data transmission channels c 12 , c 101 , and c 35 were chosen to show the process of random data loss, as shown in Figure 3. Figure 4 shows the trajectories of observation errors e in,k (i = 1, 2, . . . , 10; n = 1, 2, 3) in Example 1. It can be observed that all of the observation errors converged to zero after the step k = 4, that is, the observer statesx i,k approached the network states x i,k , which implies the lost data was compensated without bias. Along with Figure 3, it can be said that the influence of random data loss was eliminated, even if there were still data losses after k = 4.

Example 2.
A state-coupling BA scale-free network.
Consider a state-coupling BA scale-free network with 10 nodes shown in Figure 5. The corresponding network coupling configuration matrix is Three data transmission channels c 24 , c 59 , and c 82 were chosen to show the process of random data loss in Figure 6. Along with the trajectories of observation errors e in,k (i = 1, 2, . . . , 10; n = 1, 2, 3) in Example 2, which are shown in Figure 7, it can be observed that the observation errors all converged to zero after step k = 4, that is, the observer statesx i,k approached the network states x i,k , which implies the lost data was compensated without bias and the influence of random data loss was eliminated, even if there were still data losses after k = 4.
The simulation examples above cover the networks of the WS small-world and the BA scale-free topology structures, indicating that the proposed modeling method is effective in eliminating the influence of random data loss in complex dynamical networks.

Conclusions
In this paper, we proposed a new model for discrete-time complex dynamical networks with random data losses which may occur in the links between every two neighbor nodes. The data losses are described as a set of random variables satisfying a Bernoulli distribution. To construct the proposed model, an auxiliary observer was introduced, and we chose the observer states to compensate for the lost states in the coupling term. According to the analysis, we derived a sufficient condition to guarantee the compensation values finally equal to the lost values, thus the proposed model finally succeeded in eliminating the influence of data loss. From the simulation results, the proposed model was demonstrated to be effective.
In this paper, we consider only the internal data loss happening on the interaction topology in complex dynamical networks. However, it is possible to have data loss in the transmission of output variables from the observed network to the controller in the observer simultaneously. Further study could focus on ways to solve this problem. It is also interesting to mention that a number of systems and networks possess multiple time scales [39][40][41][42], where the data loss could happen on different time scales. To extend our model to complex dynamical networks where different node systems have different time scales is thus an interesting research topic. In addition, we only focused on the asymptotic convergence of the error system, and we will study the finite-time and the fixed-time asymptotic convergence in future work.