In practical applications, power grids [1, 2], communication networks, secure communication [3] and public transportation systems [4, 5] often encounter failures and attacks [6–9]. A failure of a very small fraction of nodes in a network may lead to complete fragmentation of the whole network. Therefore, the robustness of complex networks subjected to failures or attacks is an important issue to study in network science and engineering [10]. Exploring the network robustness can help better understand various networked systems and enable us to design more robust infrastructural or social systems.

In the past 2 decades, the issue of network robustness has attracted a lot of attention [10–15]. Most of the previous works focus on the structural robustness of complex networks, which is defined as the ability to maintaining their functionalities when they are disturbed or attacked [16, 17]. Therein, the failure of a node or an edge in the network is modeled by the removal of the node or the edge. To quantify structural robustness of complex networks, many measures with respect to the network structure have been formulated, such as connectivity, maximum strongly connected subgraph, natural connectivity, and average shortest path [18, 19]. Moreover, a variety of attack strategies, such as random attack and deliberate attack, have been proposed to test the robustness of different kinds of networks [20–22]. The structural robustness of complex networks can also be measured using some metrics derived from statistical physics and percolation theory [23].

Recently, the network robustness with respect to the system dynamics has stimulated even more interest [24–26]. A new concept of dynamical robustness [27] can be used to quantify the ability of a network to maintain its dynamical activities against local perturbations. Different from structural robustness where topological perturbations are considered, dynamical robustness is concerned with the robustness of network dynamics. In [27], node failure is modeled as the inactivation of diffusively coupled oscillators. In [28], dynamical robustness is quantified through the synchronization error as a function of the noise variance, where node failure is modeled by injecting noise into a node. In [29], a mathematical framework is established to quantify the effect of noise injected at one of the nodes on the synchronization performance of coupled dynamical systems. Indeed, noise is inevitable in real-world networks [30, 31]. It is natural to ask whether a networked system subjected to noise can recover to its synchronous state? On the one hand, noise may destroy the network’s stability and prevent network synchronization [32, 33]. On the other hand, noise-induced synchronization can be beneficial for coupled chaotic systems [34]. In order to clarify the influence of noise on the network, in [35] networks of different structure and complexity are analyzed, showing that many networks are better in coping with both intrinsic and extrinsic noise.

Motivated by the above discussions, this article further investigates the dynamical robustness of stochastic complex networks. The network topology is directed and the nodes are higher-dimensional non-linear dynamical systems. Similar to [29], the failure of a node in the network is modeled by injecting noise into the node. However, differing from [29], in this article it is not assumed that the network is symmetric. From a technical perspective, this introduces more challenges than its undirected counterpart [36]. A novel metric measuring the dynamical robustness of a directed networked system with additive noise is formulated. Notice that the proposed robustness metric uncovers the complex interplay between node dynamics and network topology on the overall network robustness.

The main contributions of this article are as follows. First, a mathematical framework is established to examine the dynamical robustness of a directed network of coupled dynamical systems. In this article, the notion of dynamical robustness refers to the ability of a network of coupled dynamical systems to return to its synchronous state when it encounters the disturbance of noise. The new metric is used to characterize the degree to which the networked system withstand failures and perturbations. Assume that the networked system synchronizes before the noise is introduced. The system’s robustness is defined related to the synchronization error of the network. Moreover, different from the methods used for undirected networks, the Laplacian matrix of a directed network is decomposed to two simpler matrices. In the context of mean-square stochastic stability, the new robustness metric is precisely formulated. This metric highlights the importance of the node dynamics in network robustness. Finally, numerical simulations are presented for illustration and verification using three chaotic systems (namely, Rössler system, Chen system and Wang system). The study of dynamical robustness can help better understand the roles of node dynamics and network topology, thereby better designing noise-tolerant networks.

The remainder of this article is organized as follows. Section 2 introduces the notation and some basic graph theory. In Section 3, problem formulation is presented and a new robustness metric is formulated. In Section 4, numerical simulations are shown for illustration and verification. Section 5 concludes the investigation.

Consider a directed network consisting of N identical nodes with linearly diffusive couplings, in which each node is an n-dimensional dynamical system. The network can be described by the following coupled stochastic differential equation: x ̇ i t = f x i t − α ∑ j = 1 N l i j h x j t + υ i H η η t , i = 1 , … , N , (3) where x i ( t ) ∈ R n is the state vector of node i, f (⋅) a smooth function describing the self-dynamics of each node, α > 0 the coupling strength, l _ij the (i, j)th entry of the Laplacian matrix, and h (⋅) the inner-coupling function of the nodes. The variable υ _i indicates whether node i is subjected to noise. That is, υ _i = 1 when node i is contaminated with additive noise, and υ _i = 0 otherwise. The vector H η ∈ R n describes how the noise η(t) enters the dynamics of a node, where η(t) is a zero-mean Gaussian white noise with variance θ 2 (θ > 0).

For network (3), L = ( l i j ) ∈ R N × N is the Laplacian matrix defined by l _ij = −1 if there is a directed edge from node j to node i, and l _ij = 0 otherwise, with l i i = − ∑ j = 1 , j ≠ i N l i j , for all i, j = 1, … , N. Therefore, L is a zero row-sum matrix. Recall from Section 2.2 that L = U + Δ , in which U = 1 2 ( L + L T ) is a symmetric matrix and Δ = 1 2 ( L − L T ) is an anti-symmetric matrix with Δ ^T = −Δ. Therefore, network (3) can be rewritten as x ̇ i t = f x i t − α ∑ j = 1 N u i j h x j t − α ∑ j = 1 N δ i j h x j t + υ i H η η t , i = 1 , … , N , (4) where u _ij is the (i, j)th element of U and δ _ij is the (i, j)th element of Δ.

The network is said to achieve synchronization if lim t → ∞ x i ( t ) − s ( t ) = 0 for all i = 1, … , N, where s(t) is the solution of s ̇ ( t ) = f ( s ( t ) ) (see [37]).

Define the node synchronization error ξ _i (t) = x _i (t) − s(t). The linearized error system is given by ξ ̇ i t = J f s ξ i t − α ∑ j = 1 N u i j J h s ξ j t − α ∑ j = 1 N δ i j J h s ξ j t + υ i H η η t , i = 1 , … , N , (5) where J f ( s ) ∈ R n × n and J h ( s ) ∈ R n × n are, respectively, the Jacobian matrices of f and h, i.e., J f ( s ) = ∂ f ( x ) ∂ x | x = s ( t ) and J h ( s ) = ∂ h ( x ) ∂ x | x = s ( t ) .

Let ξ ( t ) = [ ξ 1 T ( t ) , … , ξ N T ( t ) ] T ∈ R N n . The linearized error system 5) can be rewritten in a compact form as ξ ̇ t = I N ⊗ J f s − α U ⊗ J h s − α Δ ⊗ J h s ξ t + υ ⊗ H η η t , (6) where υ = [ υ 1 , … , υ N ] T ∈ R N denotes the location of the node where noise is injected.

The objective is to provide a thorough analysis of (6) to illustrate the overall effect of noise on network synchronization. Define the network synchronization error as ϕ t = ξ t 2 . (7) The expected value of ϕ(t) is given by E ϕ t = E ξ T t ξ t = Tr E ξ t ξ T t . (8) Let Σ ( t ) = E [ ξ ( t ) ξ T ( t ) ] . Note that the analysis of the effect of noise injected at the network node can be reduced to the study of the time evolution of the trace of the correlation matrix Σ(t). Since U ^T = U and Δ ^T = −Δ, one has Σ ̇ t = E ξ ̇ t ξ T t + ξ t ξ ̇ T t = I N ⊗ J f s − α U ⊗ J h s − α Δ ⊗ J h s Σ t + Σ t I N ⊗ J f T s − α U ⊗ J h T s + α Δ ⊗ J h T s + υ ⊗ H η E η t ξ T t + E ξ t η t υ T ⊗ H η T . (9)

Notice that the solution of (6) is ξ t = Φ ξ t , 0 ξ 0 + ∫ 0 t Φ ξ t , τ υ ⊗ H η η τ d τ , (10) where Φ _ξ (t, τ) is the state transition matrix associated with the state matrix I _N ⊗ J _f (s) − αU ⊗ J _h (s) − αΔ ⊗ J _h (s), and ξ(0) is the initial value. Recall that η(t) is a zero-mean Gaussian white noise with variance θ 2 . According to the analysis in [38], E η ( t ) η ( τ ) = θ 2 δ ( t − τ ) and E ξ ( 0 ) η ( τ ) = θ 2 1 N n , where δ(t) is the Dirac delta function. Eq. 9 can be rewritten as the following time-varying Lyapunov equation for the time evolution of the correlation matrix: Σ ̇ t = I N ⊗ J f s − α U ⊗ J h s − α Δ ⊗ J h s Σ t + Σ t I N ⊗ J f T s − α U ⊗ J h T s + α Δ ⊗ J h T s + θ υ υ T ⊗ H η H η T . (11)

Recall the definition of the matrix C in Lemma 1. Particularly, let C = [c ₁, … , c _N ], c i ∈ R N . Let D = C ^T ΔC, M = C ^T UC, C ̃ = C ⊗ I n , and Σ ̃ ( t ) = C ̃ T Σ ( t ) C ̃ . Multiplying (11) from the left by C ̃ T and from the right by C ̃ leads to Σ ̃ ̇ t = I N ⊗ J f s − α M ⊗ J h s − α D ⊗ J h s Σ ̃ t + Σ ̃ t I N ⊗ J f T s − α M ⊗ J h T s + α D ⊗ J h T s + θ C T υ υ T C ⊗ H η H η T . (12)

Since the trace of a matrix does not change under a similarity transformation, one has Tr ( Σ ( t ) ) = Tr ( Σ ̃ ( t ) ) and E ϕ ( t ) = Tr ( Σ ̃ ( t ) ) . Let Σ ̃ t = ∑ i , j = 1 N e i e j T ⊗ σ ̃ i j t , (13) where e i ∈ R N denotes the ith canonical vector and σ ̃ i j ( t ) ∈ R n × n is the (i, j)th block of the matrix Σ ̃ ( t ) . It then follows that E ϕ t = ∑ i = 1 N Tr σ ̃ i i t . (14) The dynamics of σ ̃ i i ( t ) are given by σ ̃ ̇ i i t = J f s − α m i i J h s σ ̃ i i t + σ ̃ i i t J f T s − α m i i J h T s + θ c i T υ 2 H η H η T , i = 1 , … , N , (15) where m _ii is the ith diagonal element of M.

Let ζ i ( t ) = σ ̃ i i ( t ) , κ _i = αm _ii , and β i = c i T υ . Eq. 15 can be rewritten as ζ ̇ i t = J f s − κ i J h s ζ i t + ζ i t J f T s − κ i J h T s + θ β i 2 H η H η T , i = 1 , … , N . (16)

Rewrite (16) as follows: V e c ζ ̇ i t = J f s − κ i J h s ⊕ J f s − κ i J h s × V e c ζ i t + θ β i 2 V e c H η H η T , i = 1 , … , N , (17) where Vec indicates matrix vectorization defined in Section 2.1. The solution of (17) is given by V e c ζ i t = Φ V e c ζ i t , 0 V e c ζ i 0 + θ β i 2 ∫ 0 t Φ V e c ζ i t , τ d τ V e c H η H η T , i = 1 , … , N , (18) where Φ V e c ( ζ i ) ( t , τ ) is the state transition matrix, which is associated with the state matrix J f ( s ) − κ i J h ( s ) ⊕ J f ( s ) − κ i J h ( s ) , and Vec (ζ _i (0)) is the initial value.

Based on (14)–(18), the expectation of the network synchronization error can be rewritten as E ϕ t = ∑ i = 1 N V e c T I n Φ V e c ζ i t , 0 V e c ζ i 0 + ∑ i = 1 N V e c T I n θ β i 2 ∫ 0 t Φ V e c ζ i t , τ d τ V e c H η H η T . (19)

In the following, consider the stochastic linear system (6), where J _f (s) and J _h (s) are Jacobian matrics of f and h evaluated at s(t), respectively. Constant α > 0 is the coupling strength and υ ∈ R N denotes the location of the node where noise is injected. The vector H η ∈ R n describes how the noise η(t) enters the dynamics of a node, where η(t) represents zero-mean Gaussian white noise with variance θ 2 . Assume that one of the network nodes denoted by i _noise is contaminated with additive noise. Let ρ = V e c T ( I n ) V e c ( ζ i noise ( t ) ) be the measurement metric of the system error, which is referred to as the robustness metric.

Remark 1. It follows from the above analysis that the robustness of the directed network subjected to noise is quantified by the robustness metric ρ . Here, the networked system synchronizes before noise is introduced. The notion of dynamical robustness refers to the ability of a network of coupled dynamical systems to return to its synchronous state after it encountered the disturbance of noise. The robustness metric ρ is used to characterize the degree to which the networked system withstand failures and perturbations. It is thus defined related to the synchronization error of the network. The smaller the value of ρ , the more robust the network. Furthermore, given the location of the node where noise is injected, the dynamical robustness of the network depends not only on the node dynamics but also on the network topology. In particular, it is determined by the inherent dynamics of the isolated node J _f (s) , the inner-coupling function J _h (s) , the coupling strength α , the variance θ of the noise, and the network topology. Note that J _f (s) and J _h (s) are the Jacobian matrices of f and h evaluated at s(t) , respectively. This implies that the robustness matric ρ is also determined by the synchronization trajectory s(t) .

In this section, the effects of node dynamics and network topology on the system robustness are investigated in detail.

This article investigates the dynamical robustness of a directed network with noise. A novel robustness metric is formulated and analyzed under the framework of mean-square stochastic stability. It is found that the dynamical robustness of the directed network is determined by both the node dynamics and the network topology. Particularly, for networks of Rössler systems, with the increase of the coupling strength, different network topologies show different effects on the robustness metric. While for Wang systems, the robustness to noise is stronger than other systems. These findings demonstrate that node dynamics plays an important role in the network robustness. The results of this study can provide theoretical and technical guidances for designing a dynamically robust networked system.

In the future, it will be interesting to investigate dynamical robustness of higher-order networks [37, 43]. Moreover, it will be interesting to investigate the effects of different types of noise on the network robustness. The dynamical robustness of stochastic complex networks with time-delays [44] or heterogeneous node dynamics [45] are challenging but also worthy of further investigation.