aSchool of Artificial Intelligence and Automation & The MOE Engineering Research Center of Autonomous Intelligent Unmanned Systems & the Key Laboratory of Image Processing, Huazhong University of Science and Technology, Wuhan 430074, China
bCenter for Polymer Studies, Department of Physics, Boston University, Boston, MA 02215, USA
Complex networked systems, which range from biological systems in the natural world to infrastructure systems in the human-made world, can exhibit spontaneous recovery after a failure; for example, a brain may spontaneously return to normal after a seizure, and traffic flow can become smooth again after a jam. Previous studies on the spontaneous recovery of dynamical networks have been limited to undirected networks. However, most real-world networks are directed. To fill this gap, we build a model in which nodes may alternately fail and recover, and we develop a theoretical tool to analyze the recovery properties of directed dynamical networks. We find that the tool can accurately predict the final fraction of active nodes, and the prediction accuracy decreases as the fraction of bidirectional links in the network increases, which emphasizes the importance of directionality in network dynamics. Due to different initial states, directed dynamical networks may show alternative stable states under the same control parameter, exhibiting hysteresis behavior. In addition, for networks with finite sizes, the fraction of active nodes may jump back and forth between high and low states, mimicking repetitive failure-recovery processes. These findings could help clarify the system recovery mechanism and enable better design of networked systems with high resilience.
Ranging from biology and ecology in the natural world to the economy and infrastructure in the human-made world, many systems can be modeled as complex networks [1]. In the past two decades, network science has provided numerous powerful tools for helping us understand the properties and functions of real networks [2], [3], [4], and network resilience is a property that has attracted much attention [5], [6], [7], [8]. Network resilience is a quality that enables a system to adjust its activity to retain its basic functionality when errors and failures occur [9]. Because networks are constantly affected by internal errors and external failures, many studies have been carried out to explore how networks lose resilience and show abrupt phase transitions from functional to collapsed [8], [10], [11], [12], [13], [14], [15]. However, in the real world, systems may spontaneously recover from collapse. For example, a patient can spontaneously recover after an epileptic attack [16], and a financial network may recover after a financial crisis [17].
To model spontaneous recovery phenomena, Majdandzic et al. [18] introduced a recovery mechanism in undirected dynamical networks and developed a framework to analyze the failure and recovery process in networks, based on the Watts model [19]. In single undirected networks, the failure-recovery mechanism at the node level can lead to macroscopic phase-flipping phenomena at the network level, which mimics the repeated inactive-active state transitions in a financial market [18], [20]. In interacting dynamical networks, a rich phase diagram with multiple triple points appears; triple points have been found to play the dominant role in constructing an optimal repair strategy in damaged interacting systems [21]. These findings have led to increased interest in network recovery [22], [23], [24]. For example, Gong et al. [25] analyzed the cascading failure phenomenon in coupled networks during recovery and proposed a recovery robustness metric to evaluate the recovery ability of coupled networks against cascading failures during recovery. Danziger and Barabasi [26] proposed a theoretical framework for solving the restorative coupling problem based on millions of recovery time data after grid failures. These studies provide theoretical support and realistic validation for spontaneous recovery in dynamic networks.
The abovementioned studies focus on undirected networks. Real-world networks, on the other hand, have directionality. Examples include metabolic networks and gene regulatory networks in biological systems [13], [27], transportation networks and power grids in infrastructure systems [28], [29], and citation networks and trust networks in social systems [30]. The states and properties of networks are largely influenced by directionality. Ignoring the directionality of the links may lead to deviations or incomplete understanding. For example, interdependent directed Erdös-Rényi (ER) networks exhibit a hybrid phase transition when a certain fraction of nodes are removed, which is not found in interdependent undirected ER networks [31], [32]. However, the existing frameworks for analyzing recovery in networks cannot be used in directed networks. An adequate theoretical analysis of recovery in directed dynamical networks is still needed.
In this work, we develop a framework for analyzing recovery in directed dynamical networks, in which nodes can fail and recover repeatedly. We find that the network state characterized by the fraction of the active nodes can be accurately predicted by our analytical framework, and it performs better than in the undirected case. The undirected case, as studied in Ref. [18], is equivalent to a special case of directed networks that have only bidirectional links. We further find that, as the fraction of bidirectional links increases, the prediction accuracy decreases because the randomness in the “transmitter-infected” relationships decreases as the fraction of bidirectional links increases. By applying the framework to synthesis networks, we find that strong hysteresis and first-order phase transitions appear in large-scale networks, and the network state-flipping phenomenon appears in finite-scale directed networks. In addition, we analyze the influence of network structure and dynamical parameters on network resilience and find that the larger the propagation range, the worse the recoverability of the system.
2. Modeling spontaneous recovery in directed dynamical networks
We build a model to describe the process of spontaneous recovery in directed dynamical networks. Given a directed network comprising N nodes, every node is assigned a state: active or failed. A node may fail internally or externally, based on the following assumptions: ① Every node in the directed network can fail with a probability p during a time interval and is not affected by other nodes, which is called internal failure (Fig. 1(a)); and ② every node may fail with a probability of r (Fig. 1(b)) because the fraction of its active incoming neighbors among all incoming neighbors is lower than an external influence threshold mf∈[0,1], which describes the degree to which a node is affected by its incoming neighbors. The probability r, called the external failure probability, characterizes the rate at which failures propagate through the network. The thresholds mf and r influence the external failure of the network.
Any failed nodes can recover from internal failures after a time period of τ ≠ 0. That is, if a node fails due to internal failure at time t, it will recover at t + τ. In addition, any node will recover from an external failure after a time period τ'. For the simplicity of subsequent analysis, we set τ' = 1 (Figs. 1(c) and (d)). Based on these assumptions, the nodes may switch back and forth between the active and failed states. The fraction of active node at time t is denoted as z.
3. Developing an analytical tool to calculate the fraction of active nodes
We developed a theoretical tool to analyze the recovery properties of directed dynamical networks based on mean field theory (MFT). This tool can be used to predict the fraction of active nodes in a directed dynamical network.
A node is said to be in a critically damaged neighborhood if the fraction of active incoming neighbors is less than mf. The parameter Ekin denotes the probability that a node with an in-degree of kin is in a critically damaged neighborhood. The average fraction of failed nodes is denoted as a (0 < a < 1). According to combinatorics theory, it is possible to obtain the relation between Ekin and a:
where j is the number of the active incoming neighbor nodes.
Furthermore, the probability of the external failure of a node with an in-degree of kin is rEkin. The probability of a node’s internal failure is related to the product of p and τ, where p∈[0,1] is the probability that a node internally fails during a unit interval τ. Therefore, we define a variable $ p^{*}=1-\exp (-p \tau)$ to represent the time-averaged percentage of nodes in failed states due to internal failure.
We assume that internal and external failures are independent. The probability of failure of a node with internal degree kin is $ a k_{\mathrm{in}}=p^{*}+r E_{k_{\mathrm{in}}}-p^{*} r E_{k_{\mathrm{in}}}$. Summing over all the nodes, the fraction of nodes in the network that are in failed states is obtained, as follows:
where P(kin) denotes the probability that the in-degree of a node is kin. It should be noted that, as shown in Eq. (2), the fraction of failed nodes is related to the nodes’ in-degrees; this is reasonable since the states of the upstream nodes can affect a node’s state. However, this does not mean that the node’s out-degrees have no influence on the network state. In experiments where nodes have different recovery rates, we find that increasing the recovery rates of the nodes with high out-degrees can promote the recovery of the whole network (Section S1 and Figs. S1−S3 in Appendix A for further details).
The average percentage of failed nodes can be obtained by solving Eq. (2). Under certain values of r and p*, Eq. (2) may have one solution. For some other values of r and p*, the equation may have three solutions, with the highest and the lowest being stable states and the middle solution being an unstable state. The average fraction of active nodes is denoted as $ \langle z\rangle=\left\langle z\left(r, p^{*}\right)\right\rangle=1-a$, which also represents the network state.
4. Hysteresis in large-scale networks
We apply the analytical tool to ER networks and find that large-scale directed dynamical networks may show strong hysteresis. In a directed ER network with an average degree $ \langle k\rangle$, the probability of the in-degree of a randomly chosen node being kin is P(kin):
where p represents the probability that any two nodes are connected, which is related to the average degree $ \langle k\rangle$ of the directed network:
$ p=\frac{\langle k\rangle}{2(N-1)}$
For ER networks composed of a large number of nodes N, the degree has a Poisson distribution. For a directed ER network with an average degree of $ \langle k\rangle$, the fraction of failed nodes is:
The dynamical network reaches a dynamical equilibrium after a certain time.
Next, we study how the network state changes under different internal and external failures. As p* increases, $\langle z\rangle$ decreases and shows different phase transitions under different external failure probabilities r (Fig. 2(a)). When r is small, $\langle z\rangle$ continuously decreases to zero, showing a second-order phase transition. In such cases, there is a single solution for $\langle z\rangle$. If we reduce p*, $\langle z\rangle$ will increase through the same curve as the direction of increasing p*. When r is large, the network state shows an abrupt first-order phase transition. Under certain values of p* and r, there are three solutions for $\langle z\rangle$, with the highest and the lowest being stable states and the middle solution being an unstable state. In the direction of increasing p*, $\langle z\rangle$ decreases and abruptly jumps to a small value at a critical point p2∗. In the direction of decreasing p*, $\langle z\rangle$ discontinuously increases to a large value at another critical point p1∗. In a highly active state, many nodes are active, which makes it easy to externally recover the failed nodes. In a low-activity state, many nodes fail, and external failure easily spreads. Thus, the critical point is p2∗>p1∗, yielding hysteresis. Similar results are obtained when we change mf (Section S2 and Fig. S4 in Appendix A).
We next draw a phase diagram in the r−p* space. As shown in Fig. 2(b), the two black curves represent the two critical points p1∗ and p2∗ for different p* and r. The space is divided into three regions by the two curves: ① a high-activity region (region A); ② a low-activity region (region B); and ③ a hysteresis region. In the hysteresis region, $\langle z\rangle$ can either be high or low, depending on the direction of changing p*. As r increases, the critical points p* decrease, and one critical point p1∗ disappears at a certain point, which means that the network cannot be recovered to a high-activity state even when p* = 0. In addition, the solid lines in Fig. 2 represent the analytical solutions of our theoretical framework, which agree well with the simulation results, represented by the dots.
Whether hysteresis appears is not only determined by p* and r but also influenced by mf, as mf and r together determine the propagation of external failure. We further draw a three-dimensional (3D) figure to show when hysteresis appears in the mf−r−p* space. When mf is too large (mf > 0.8) or too small (mf < 0.15), there is no hysteresis (Fig. 2(c)). If mf is too large, the failure will rapidly propagate to other nodes, and the network will quickly reach a low-activity equilibrium. If mf is too small, the network will reach a high-activity equilibrium. The appearance of hysteresis is determined by a small internal failure probability and moderate external failure propagation. It should be noted that the 3D phase diagram is not perfectly smooth; this is because the degrees of the nodes are integers and not uniformly distributed.
It is also notable that the network state in our work is defined by the average percentage of active nodes. In many previous studies on network resilience, the size of the giant connected component (GCC) has been used to represent the functional component of a network. We also study network recovery by using the size of the GCC as a metric to define the network state, and we find that the recovery properties are similar. The critical points where the first-order phase transition occurs are the same as in the case where the network state is defined by the average percentage of active nodes (see Section S2 in Appendix A for further details).
5. Flipping phenomenon in dynamical networks with finite sizes
In large-scale networks, hysteresis appears due to different initial network states. That is, if the network is initially in a low-activity state, the network will stay in a low-activity state after a repetitive failure-recovery process; if it starts in a high-activity state, it will end up in a high-activity state. In contrast, in a small-scale network, we find that the network may flip between high-activity and low-activity states.
As shown in Fig. 3, in a finite-scale ER network with 50 nodes, z flips between two regions, which are centered at zhigh≈0.89 and zlow≈0.23. The state-flipping phenomenon appears due to the randomness in the configuration of active and failed nodes. Certain configurations with active nodes that have higher degrees will rapidly recover most nodes, driving the network into a high-activity state, and configurations with failed high-degree nodes will spread failure quickly and drive the network into a low-activity state. It should be noted that the state-flipping phenomenon does not always appear in finite networks. When the dynamical parameters are set as in the hysteresis region of large-scale networks, the flipping phenomenon is more common than in other regions.
Furthermore, we have identified the occurrence of the state-flipping phenomenon in random regular (RR) and scale-free (SF) networks, as depicted in Fig. 3. To elucidate the impact of network structural properties on this phenomenon, we conducted experiments in networks with varying average degrees and degree distribution exponents (see Figs. S5−S8 in Appendix A for further details). We observed that, in a network characterized by a low average degree, the network predominantly resides in a state of low activity, and the state-flipping phenomenon may not manifest itself prominently. As the network density increases, the state-flipping phenomenon becomes more pronounced, leading to an extended duration during which the network remains in a high-activity state. This observation suggests that a higher average degree can promote a greater degree of spontaneous recovery within the network (see Figs. S9 and S10 in Appendix A for further details). In SF networks, as the degree distribution exponent increases, indicating a decrease in degree heterogeneity, the duration for which a network remains in a high-activity state diminishes. This suggests that enhancing degree heterogeneity can facilitate recovery within the network (see Fig. S11 in Appendix A for further details).
In small-scale networks, the network state may flip between high-activity and low-activity states, and z may show oscillation as the fraction of internally failed nodes increases, as shown in Fig. 4. This oscillation is caused by the randomness in the configurations of active and failed nodes. We draw a phase diagram of the network states in the space of p* and r, which is composed of a high-activity region, a low-activity region, and an oscillation region. As the network scale increases, the oscillation amplitude decreases, and the oscillation region in the phase diagram decreases. When the network scale is large enough, the oscillation region disappears. That is, the influence of the randomness caused by different configurations of active and failed nodes decreases as the network scale increases and disappears for infinite networks.
6. Comparison of spontaneous recovery in directed versus undirected networks
As shown in Fig. 2(b), the critical points calculated by using our analytical framework (black solid lines) agree very well with the simulations (white dots). For the framework developed for undirected networks [18], there is a nonnegligible gap between the theoretical result and the simulation. This is because the mean field method assumes that the networks are infinite while the networks for the simulation are finite; we also find that the symmetry in “transmitter-infected” relationships has a great impact on the agreement between theory and simulations.
MFT usually applies to random models. In the recovery model, failure or recovery spreads via “transmitter-infected” relationships. In undirected networks, the “transmitter-infected” relationships are perfectly symmetric and not random. For every node pair “A—B”, node A is a transmitter for node B, and node B is also a transmitter for node A. In contrast, in an ordinary directed network, this relationship is close to random. An undirected network can be treated as a special case of a directed network, with every link being bidirectional. In an ordinary directed network, the fraction of bidirected links is usually low. Thus, we hypothesize that, as the fraction of bidirectional links increases, the accuracy of the analytical framework decreases.
To verify this hypothesis, we keep the degree of each node of a directed network and change the fraction of bidirectional links b. According to the theoretical tool, since both the in-degree and out-degree of the network remain unchanged, the theoretical values remain the same. The simulation curves of z shift to the right as b increases (Fig. 5(a)). This indicates that the symmetry in “transmitter-infected” relationships could help improve network activity. We further draw the r−p* phase diagram and find that the gap between the theoretical results and the simulation increases as the fraction of bidirectional links b increases, verifying our assumption (Fig. 5(b)), which is because the randomness in the “transmitter-infected” relationships decreases as b increases. In addition, we find that the difference in the network activity caused by different fractions of bidirectional links b decreases as r increases to a large enough value, and the critical points in networks with different r coincide near r = 0.9935 (Fig. 5(c)). If r continues to increase, the differences between network activity may reverse, with the critical points in networks with larger b becoming smaller than those in networks with smaller b.
Moreover, we apply the failure-recovery model to SF and RR networks and find similar results as those found in ER networks (see Section S3 in Appendix A for further details). In SF networks, as the degree distribution exponent increases, z decreases, which indicates that the degree heterogeneity could help increase the network activity.
7. Conclusions
When nodes or connections fail in a network, many systems in the real world may exhibit spontaneous recovery, which can be seen as an important manifestation of system resilience. Previous research on the spontaneous recovery of dynamic networks has mainly focused on undirected networks, while many networks in the real world are directed. To fill this gap, we established a model in which nodes can alternately fail and recover. Furthermore, we developed a theoretical tool to analyze the spontaneous recovery of directed dynamical networks and demonstrate the tool’s predictive power for the fraction of final active nodes in the network. We found that the tool can accurately predict the fraction of final active nodes, and the prediction accuracy decreases as the fraction of bidirectional links in the network increases, which emphasizes the importance of directionality in network dynamics. In addition, we find that directed dynamical networks may exhibit alternative stable states and hysteresis behavior under the same control parameter.
Our findings have implications for understanding the mechanisms of system recovery and for designing resilient networked systems in various domains, such as transportation, communication, and biological systems. For example, the results suggest that increasing the fraction of bidirectional links in a network enhances its resilience to failure. Moreover, the observation of alternative stable states and hysteresis behavior in directed dynamical networks implies that interventions to recover a network from failure may require careful consideration of the initial conditions and control parameters.
The model in our work is non-Markov recovery, in which a failed node will recover after a certain period (e.g., τ = 100). We compared this to Markov recovery, in which every failed node has a certain probability (e.g., 1/τ = 0.01) of recovering at each simulation step, and found that a network with Markov recovery has a higher active state than a network with non-Markov recovery when the other parameters are the same; moreover, the critical point where the first-order phase transition occurs is larger. This indicates that Markov recovery could increase the network recovery, which is reasonable since a failed node can recover at each step in Markov recovery, and the recovery of this node can trigger more nodes to recover (Fig. S12 in Appendix A).
Future research directions include the study of the effects of different types of directional links—such as inhibitory and excitatory links—on the network’s recovery properties, as well as studies on real networks. The recovery process in real networks may lead to different properties, such as the transitions between states in hysteresis being gentler than those of synthesis networks (Fig. S13 in Appendix A). In addition, networks may have higher-order structures that markedly affect the dynamics [33], [34], [35], [36]. The question of how higher-order network structures could affect network recovery is worth investigating in the future. Furthermore, the experimental validation of our theoretical predictions can provide valuable insights into the recovery mechanisms of real-world directed dynamical networks. Ultimately, a better understanding of the recovery properties of directed dynamical networks can inform the design and management of resilient networked systems in various domains.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (62172170) and the Science and Technology Project of the State Grid Corporation of China (5100-202199557A-0-5-ZN).
Compliance with ethics guidelines
Xueming Liu, Xian Yan, and H. Eugene Stanley declare that they have no conflict of interest or financial conflicts to disclose.
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