Optimal weighting scheme for suppressing cascades and traffic congestion in complex networks

Research on proactive defense and dynamic repair of complex networks considering cascading effects

Cascading effects can result in the nonlinear propagation of failures in complex networks, ultimately leading to network collapse. Research on the fault propagation principles, defense strategies, and repair strategies can help mitigate the effects of cascading failures. Especially, proactive defense and dynamic repair are flexible and effective methods to ensure network security. Most studies on the cascade of complex networks are based on the unprocessed initial information of the network. However, marginal nodes are a type of node that cloaks the initial information of the network. In this study, we rank the importance of nodes according to the intensity of network energy confusion after the removal of this node, clarify the meaning of marginal nodes and proposed two methods to screen marginal nodes. The results indicated that the proactive removal of marginal nodes can effectively reduce the effect of cascading failures without causing any negative disturbance to the energy flow of the network. In addition, network repair according to the proposed strategy can minimize the cascade effect in the repair process and improve repair efficiency.

MLCOR Model for Suppressing the Cascade of Edge Failures in Complex Network

As a decisive parameter of network robustness and network economy, the capacity of network edges can directly affect the operation stability and the construction cost of the network. This paper proposes a multilevel load–capacity optimal relationship (MLCOR) model that can substantially improve the network economy on the premise of network safety. The model is verified in artificially created networks including free-scale networks, small-world networks, and in the real network structure of the Shanghai Metro network as well. By numerical simulation, it is revealed that under the premise of ensuring the stability of the network from the destruction caused by initial internal or external damage on edge, the MLCOR model can effectively reduce the cost of the entire network compared to the other two linear load–capacity models regardless of what extent of the destruction that the network edges suffer initially. It is also proved that there exists an optimal tunable parameter and the corresponding optimal network cost for any BA and NW network topology, which can provide the reference for setting reasonable capacities for network edges in a real network at the stage of network planning and construction, promoting security and stability of network operation.

Modelling cascading failures in networks with the harmonic closeness

Many studies on cascading failures adopt the degree or the betweenness of a node to define its load. From a novel perspective, we propose an approach to obtain initial loads considering the harmonic closeness and the impact of neighboring nodes. Based on simulation results for different adjustable parameter θ, local parameter δ and proportion of attacked nodes f, it is found that in scale-free networks (SF networks), small-world networks (SW networks) and Erdos-Renyi networks (ER networks), there exists a negative correlation between optimal θ and δ. By the removal of the low load node, cascading failures are more likely to occur in some cases. In addition, we find a valuable result that our method yields better performance compared with other methods in SF networks with an arbitrary f, SW and ER networks with large f. Moreover, the method concerning the harmonic closeness makes these three model networks more robust for different average degrees. Finally, we perform the simulations on twenty real networks, whose results verify that our method is also effective to distribute the initial load in different real networks.

Identifying edges that facilitate the generation of extreme events in networked dynamical systems

The collective dynamics of complex networks of FitzHugh-Nagumo units exhibits rare and recurrent events of high amplitude (extreme events) that are preceded by so-called proto-events during which a certain fraction of the units become excited. Although it is well known that a sufficiently large fraction of excited units is required to turn a proto-event into an extreme event, it is not yet clear how the other units are being recruited into the final generation of an extreme event. Addressing this question and mimicking typical experimental situations, we investigate the centrality of edges in time-dependent interaction networks. We derived these networks from time series of the units' dynamics employing a widely used bivariate analysis technique. Using our recently proposed edge-centrality concepts together with an edge-based network decomposition technique, we observe that the recruitment is primarily facilitated by sets of certain edges that have no equivalent in the underlying topology. Our finding might aid to improve the understanding of generation of extreme events in natural networked dynamical systems.

Non-Markovian recovery makes complex networks more resilient against large-scale failures

Non-Markovian spontaneous recovery processes with a time delay (memory) are ubiquitous in the real world. How does the non-Markovian characteristic affect failure propagation in complex networks? We consider failures due to internal causes at the nodal level and external failures due to an adverse environment, and develop a pair approximation analysis taking into account the two-node correlation. In general, a high failure stationary state can arise, corresponding to large-scale failures that can significantly compromise the functioning of the network. We uncover a striking phenomenon: memory associated with nodal recovery can counter-intuitively make the network more resilient against large-scale failures. In natural systems, the intrinsic non-Markovian characteristic of nodal recovery may thus be one reason for their resilience. In engineering design, incorporating certain non-Markovian features into the network may be beneficial to equipping it with a strong resilient capability to resist catastrophic failures.

Asymmetry in interdependence makes a multilayer system more robust against cascading failures

Multilayer networked systems are ubiquitous in nature and engineering, and the robustness of these systems against failures is of great interest. A main line of theoretical pursuit has been percolation-induced cascading failures, where interdependence between network layers is conveniently and tacitly assumed to be symmetric. In the real world, interdependent interactions are generally asymmetric. To uncover and quantify the impact of asymmetry in interdependence on network robustness, we focus on percolation dynamics in double-layer systems and implement the following failure mechanism: Once a node in a network layer fails, the damage it can cause depends not only on its position in the layer but also on the position of its counterpart neighbor in the other layer. We find that the characteristics of the percolation transition depend on the degree of asymmetry, where the striking phenomenon of a switch in the nature of the phase transition from first to second order arises. We derive a theory to calculate the percolation transition points in both network layers, as well as the transition switching point, with strong numerical support from synthetic and empirical networks. Not only does our work shed light on the factors that determine the robustness of multilayer networks against cascading failures, but it also provides a scenario by which the system can be designed or controlled to reach a desirable level of resilience.

Threshold driven contagion on weighted networks

Weighted networks capture the structure of complex systems where interaction strength is meaningful. This information is essential to a large number of processes, such as threshold dynamics, where link weights reflect the amount of influence that neighbours have in determining a node's behaviour. Despite describing numerous cascading phenomena, such as neural firing or social contagion, the modelling of threshold dynamics on weighted networks has been largely overlooked. We fill this gap by studying a dynamical threshold model over synthetic and real weighted networks with numerical and analytical tools. We show that the time of cascade emergence depends non-monotonously on weight heterogeneities, which accelerate or decelerate the dynamics, and lead to non-trivial parameter spaces for various networks and weight distributions. Our methodology applies to arbitrary binary state processes and link properties, and may prove instrumental in understanding the role of edge heterogeneities in various natural and social phenomena.

Mobility and Congestion in Dynamical Multilayer Networks with Finite Storage Capacity

Multilayer networks describe well many real interconnected communication and transportation systems, ranging from computer networks to multimodal mobility infrastructures. Here, we introduce a model in which the nodes have a limited capacity of storing and processing the agents moving over a multilayer network, and their congestions trigger temporary faults which, in turn, dynamically affect the routing of agents seeking for uncongested paths. The study of the network performance under different layer velocities and node maximum capacities reveals the existence of delicate trade-offs between the number of served agents and their time to travel to destination. We provide analytical estimates of the optimal buffer size at which the travel time is minimum and of its dependence on the velocity and number of links at the different layers. Phenomena reminiscent of the slower is faster effect and of the Braess' paradox are observed in our dynamical multilayer setup.

The "weak" interdependence of infrastructure systems produces mixed percolation transitions in multilayer networks

Previous studies of multilayer network robustness model cascading failures via a node-to-node percolation process that assumes "strong" interdependence across layers-once a node in any layer fails, its neighbors in other layers fail immediately and completely with all links removed. This assumption is not true of real interdependent infrastructures that have emergency procedures to buffer against cascades. In this work, we consider a node-to-link failure propagation mechanism and establish "weak" interdependence across layers via a tolerance parameter α which quantifies the likelihood that a node survives when one of its interdependent neighbors fails. Analytical and numerical results show that weak interdependence produces a striking phenomenon: layers at different positions within the multilayer system experience distinct percolation transitions. Especially, layers with high super degree values percolate in an abrupt manner, while those with low super degree values exhibit both continuous and discontinuous transitions. This novel phenomenon we call mixed percolation transitions has significant implications for network robustness. Previous results that do not consider cascade tolerance and layer super degree may be under- or over-estimating the vulnerability of real systems. Moreover, our model reveals how nodal protection activities influence failure dynamics in interdependent, multilayer systems.

Controlling congestion on complex networks: fairness, efficiency and network structure

We consider two elementary (max-flow and uniform-flow) and two realistic (max-min fairness and proportional fairness) congestion control schemes, and analyse how the algorithms and network structure affect throughput, the fairness of flow allocation, and the location of bottleneck edges. The more realistic proportional fairness and max-min fairness algorithms have similar throughput, but path flow allocations are more unequal in scale-free than in random regular networks. Scale-free networks have lower throughput than their random regular counterparts in the uniform-flow algorithm, which is favoured in the complex networks literature. We show, however, that this relation is reversed on all other congestion control algorithms for a region of the parameter space given by the degree exponent γ and average degree 〈k〉. Moreover, the uniform-flow algorithm severely underestimates the network throughput of congested networks, and a rich phenomenology of path flow allocations is only present in the more realistic α-fair family of algorithms. Finally, we show that the number of paths passing through an edge characterises the location of a wide range of bottleneck edges in these algorithms. Such identification of bottlenecks could provide a bridge between the two fields of complex networks and congestion control.

Growth, collapse, and self-organized criticality in complex networks

Network growth is ubiquitous in nature (e.g., biological networks) and technological systems (e.g., modern infrastructures). To understand how certain dynamical behaviors can or cannot persist as the underlying network grows is a problem of increasing importance in complex dynamical systems as well as sustainability science and engineering. We address the question of whether a complex network of nonlinear oscillators can maintain its synchronization stability as it expands. We find that a large scale avalanche over the entire network can be triggered in the sense that the individual nodal dynamics diverges from the synchronous state in a cascading manner within a relatively short time period. In particular, after an initial stage of linear growth, the network typically evolves into a critical state where the addition of a single new node can cause a group of nodes to lose synchronization, leading to synchronization collapse for the entire network. A statistical analysis reveals that the collapse size is approximately algebraically distributed, indicating the emergence of self-organized criticality. We demonstrate the generality of the phenomenon of synchronization collapse using a variety of complex network models, and uncover the underlying dynamical mechanism through an eigenvector analysis.

Contagion processes on the static and activity-driven coupling networks

The evolution of network structure and the spreading of epidemic are common coexistent dynamical processes. In most cases, network structure is treated as either static or time-varying, supposing the whole network is observed in the same time window. In this paper, we consider the epidemics spreading on a network which has both static and time-varying structures. Meanwhile, the time-varying part and the epidemic spreading are supposed to be of the same time scale. We introduce a static and activity-driven coupling (SADC) network model to characterize the coupling between the static ("strong") structure and the dynamic ("weak") structure. Epidemic thresholds of the SIS and SIR models are studied using the SADC model both analytically and numerically under various coupling strategies, where the strong structure is of homogeneous or heterogeneous degree distribution. Theoretical thresholds obtained from the SADC model can both recover and generalize the classical results in static and time-varying networks. It is demonstrated that a weak structure might make the epidemic threshold low in homogeneous networks but high in heterogeneous cases. Furthermore, we show that the weak structure has a substantive effect on the outbreak of the epidemics. This result might be useful in designing some efficient control strategies for epidemics spreading in networks.

Opening bottlenecks on weighted networks by local adaptation to cascade failures

Structure and dynamics of complex systems are often described using weighted networks in which the position, weight and direction of links quantify how activity propagates between system elements, or nodes. Nodes with only few outgoing links of low weight have low out-strength and thus form bottlenecks that hinder propagation. It is currently not well understood how systems can overcome limits imposed by such bottlenecks. Here, we simulate activity cascades on weighted networks and show that, for any cascade length, activity initially propagates towards high out-strength nodes before terminating in low out-strength bottlenecks. Increasing the weights of links that are active early in the cascade further enhances already strong pathways, but worsens the bottlenecks thereby limiting accessibility to other pathways in the network. In contrast, strengthening only links that propagated the activity just prior to cascade termination, i.e. links that point into bottlenecks, eventually removes these bottlenecks and increases the accessibility of all paths on the network. This local adaptation rule simply relies on the relative timing to a global failure signal and allows systems to overcome engrained structure to adapt to new challenges.

Ability paradox of cascading model based on betweenness

Must Investing more resources to protect every node in a network improve the robustness of the whole network subject to target attacks? To answer this question, we investigate the cascading dynamics in some typical networks. In real networks, the load on a node is generally correlated with the betweenness. Considering the weight of a node, we give a new method to define the initial load on a node by the revised betweenness. Then we present a simple cascading model. We investigate the cascading dynamics by disabling a single key node with the highest load. We find that in BA scale-free networks, the bigger the capacity of every node, the stronger the robustness of the whole network. However, in WS networks and some random networks, when we increase the capacity of every node, instead, the robustness of the whole network is weaker. In US power grid and the China power grid, we also observe this counterintuitive phenomenon. We give a reasonable explanation by a simple illusion. By the analysis, we think that resurrections of some nodes in a ring network structure after removing a node may be the reason of this phenomenon.

Robustness of controllability for networks based on edge-attack

We study the controllability of networks in the process of cascading failures under two different attacking strategies, random and intentional attack, respectively. For the highest-load edge attack, it is found that the controllability of Erdős-Rényi network, that with moderate average degree, is less robust, whereas the Scale-free network with moderate power-law exponent shows strong robustness of controllability under the same attack strategy. The vulnerability of controllability under random and intentional attacks behave differently with the increasing of removal fraction, especially, we find that the robustness of control has important role in cascades for large removal fraction. The simulation results show that for Scale-free networks with various power-law exponents, the network has larger scale of cascades do not mean that there will be more increments of driver nodes. Meanwhile, the number of driver nodes in cascading failures is also related to the edges amount in strongly connected components.

Optimal transport on weighted networks for different node delivery capability schemes

Many real networks can be best described by weighted networks with a diversity of interactions between nodes measured by the weights of the edges. It is of great importance to improve the overall capacity of these real-world networks. In this paper, the traffic capacity of weighted network is investigated based on three different node delivery capability schemes: the delivery capacity of each node is constant in the first scheme while in the second and third schemes it is proportional to its node degree and node strength. It is shown by simulations that the network transfer capacity depends strongly on the tunable parameter. And different tunable parameter is suitable for different node delivery capability.

Resiliently evolving supply-demand networks

The ability to design a transport network such that commodities are brought from suppliers to consumers in a steady, optimal, and stable way is of great importance for distribution systems nowadays. In this work, by using the circuit laws of Kirchhoff and Ohm, we provide the exact capacities of the edges that an optimal supply-demand network should have to operate stably under perturbations, i.e., without overloading. The perturbations we consider are the evolution of the connecting topology, the decentralization of hub sources or sinks, and the intermittence of supplier and consumer characteristics. We analyze these conditions and the impact of our results, both on the current United Kingdom power-grid structure and on numerically generated evolving archetypal network topologies.

Cascading dynamics on random networks: crossover in phase transition

In a complex network, random initial attacks or failures can trigger subsequent failures in a cascading manner, which is effectively a phase transition. Recent works have demonstrated that in networks with interdependent links so that the failure of one node causes the immediate failures of all nodes connected to it by such links, both first- and second-order phase transitions can arise. Moreover, there is a crossover between the two types of transitions at a critical system-parameter value. We demonstrate that these phenomena can occur in the more general setting where no interdependent links are present. A heuristic theory is derived to estimate the crossover and phase-transition points, and a remarkable agreement with numerics is obtained.

Local empathy provides global minimization of congestion in communication networks

We present a mechanism to avoid congestion in complex networks based on a local knowledge of traffic conditions and the ability of routers to self-coordinate their dynamical behavior. In particular, routers make use of local information about traffic conditions to either reject or accept information packets from their neighbors. We show that when nodes are only aware of their own congestion state they self-organize into a hierarchical configuration that delays remarkably the onset of congestion although leading to a sharp first-order-like congestion transition. We also consider the case when nodes are aware of the congestion state of their neighbors. In this case, we show that empathy between nodes is strongly beneficial to the overall performance of the system and it is possible to achieve larger values for the critical load together with a smooth, second-order-like, transition. Finally, we show how local empathy minimize the impact of congestion as much as global minimization. Therefore, here we present an outstanding example of how local dynamical rules can optimize the system's functioning up to the levels reached using global knowledge.

Effective strategy of adding nodes and links for maximizing the traffic capacity of scale-free network

In this paper, we propose an efficient strategy to enhance traffic capacity via the process of nodes and links increment. We show that by adding shortcut links to the existing networks, packets are avoided flowing through hub nodes. We investigate the performances of our proposed strategy under the shortest path routing strategy and the local routing strategy. Our obtained results show that using the proposed strategy, the traffic capacity can be effectively enhanced under the shortest path routing strategy. Under the local routing strategy, the obtained results show that the proposed strategy is efficient only when packets are more likely to be forwarded to low-degree nodes in their routing paths. Compared with other strategies, the obtained results indicate that our proposed strategy of adding nodes and links is the most effective in enhancing the traffic capacity, i.e., the traffic capacity can be maximally enhanced with the least number of additional nodes and links.

Motif distributions in phase-space networks for characterizing experimental two-phase flow patterns with chaotic features

The dynamics of two-phase flows have been a challenging problem in nonlinear dynamics and fluid mechanics. We propose a method to characterize and distinguish patterns from inclined water-oil flow experiments based on the concept of network motifs that have found great usage in network science and systems biology. In particular, we construct from measured time series phase-space complex networks and then calculate the distribution of a set of distinct network motifs. To gain insight, we first test the approach using time series from classical chaotic systems and find a universal feature: motif distributions from different chaotic systems are generally highly heterogeneous. Our main finding is that the distributions from experimental two-phase flows tend to be heterogeneous as well, suggesting the underlying chaotic nature of the flow patterns. Calculation of the maximal Lyapunov exponent provides further support for this. Motif distributions can thus be a feasible tool to understand the dynamics of realistic two-phase flow patterns.