**2. Model of complex networks with non-delayed and delayed couplings over random exchanges**

As is known, time delay is ubiquitous in many network systems. When time delay exists in the interaction, it may affect the dynamic behavior and even destabilize the network system. Thus, time delay should be taken into consideration, which could accurately reflect some characteristics of networks. By investing the existing literatures, it is easy to find that most of the results on complex networks have been carried out under some implicit assumptions. That is the communication information of nodes is only related to *x*(*t*) or *x*(*t* − *τ*). However, in many cases, this simplification is not satisfactory for the special nature of the networks. In fact, the information communication of nodes is not only related to *x*(*t*) but also to *x*(*t* − *τ*). Unfortu‐ nately, this property has been ignored in many literatures that are about the complex systems with non-delayed and delayed couplings simultaneously. In this section, we will consider a general stabilization problem of complex systems with non-delayed and delayed couplings exchanging randomly.

Considering a kind of complex dynamical network consisting of *N* nodes and every node is a *n* -dimensional dynamical system, which is described as

$$\dot{\mathbf{x}}\_{i}(t) = f\left(\mathbf{x}\_{i}(t)\right) + c\sum\_{j=1}^{N} a\_{ij}\mathbf{x}\_{j}(t) + c\sum\_{j=1}^{N} b\_{ij}\mathbf{x}\_{j}(t-\tau), \quad i \in \mathcal{S} \tag{1}$$

where *xi* (*t*) = (*xi*1(*t*), *xi*2(*t*), …, *xin*(*t*))*<sup>T</sup>* ∈ ▯*<sup>n</sup>* is the state vector of the *i*th node. *f* : ▯*<sup>n</sup>* → ▯*<sup>n</sup>* is a continuously differentiable function that describes the activity of an individual system.

*c* > 0 is the coupling strength among the nodes. *τ* > 0 is the coupling delay. *A* = (*aij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>* and *B* = (*bij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>* stand for the configuration matrices of the complex dynamical network with the non-delayed and delayed couplings, respectively. *A* and *B* can be defined as follows: for *i* ≠ *j*, if there exist non-delayed and delayed couplings between nodes *i* and *j*, then *aij* > 0 and *bij* > 0; Otherwise, *aij* = 0 and *bij* = 0, respectively. Assuming both *A* and *B* are symmetric and also satisfy

$$a\_{\boldsymbol{\mu}} = -\sum\_{j=1, j\neq i}^{N} a\_{\boldsymbol{\mu}}, b\_{\boldsymbol{\mu}} = -\sum\_{j=1, j\neq i}^{N} b\_{\boldsymbol{\mu}}, i \in \mathbf{S}^{\boldsymbol{\mu}}$$

Here, the topologies of the complex network are more general, whose related coupling matrices exchange each other randomly. That is, *A* changes into *B*, while *B* changes into *A* simultane‐ ously. In other words, matrices *A* and *B* exchange. In this case, we have the following complex network:

$$\dot{\mathbf{x}}\_{i}(t) = f\left(\mathbf{x}\_{i}(t)\right) + c \sum\_{j=1}^{N} b\_{ij} \mathbf{x}\_{j}(t) + c \sum\_{j=1}^{N} a\_{ij} \mathbf{x}\_{j}(t - \tau), \quad i \in \mathcal{S} \tag{2}$$

From these demonstrations, it is seen that the above two complex networks occur separately and randomly. To describe the above random switching between coupling matrices *A* and *B*, a robust method will be exploited. That is

$$\dot{\mathbf{x}}\_{i}(t) = f\left(\mathbf{x}\_{i}(t)\right) + c\sum\_{j=1}^{N} (a\_{ij} + \Delta a\_{ij})\mathbf{x}\_{j}(t) + c\sum\_{j=1}^{N} (b\_{ij} + \Delta b\_{ij})\mathbf{x}\_{j}(t-\tau), \quad i \in \mathcal{S} \tag{3}$$

when *ΔA* = (*Δaij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>* and *ΔB* = (*Δbij*) ∈ ▯*<sup>N</sup>* × *<sup>N</sup>*. Especially, such uncertainties are selected to be *ΔA* = *B* − *A* and *ΔB* = *A* − *B*, which is assumed to be

$$B - A \quad \le \delta^\* \tag{4}$$

where *δ*\* is a given positive scalar.

Before giving the main results, a definition is needed.

**Definition 1.** The complex network (1) is asymptotically stable over topologies exchanging randomly, if the complex network (3) with condition (4) is asymptotically stable.
