Vertex overload breakdown in evolving networks

A Simulation Study of the Resiliency of Mobile Energy Storage Networks

Resilience is regarded as an essential design objective of a wide range of systems in modern society. This work is based on a vision that networks of mobile energy storage systems could provide an alternative off-grid power system design for rural and underdeveloped regions. To evaluate the resiliency of networked energy storage systems under overload failure, a model of concurrent cascading failure and healing processes is developed and demonstrated. Two resilience metrics are used to evaluate the resilience of a real-world network, namely the recovery level at a specified time and the recovery time. The simulations generate system trajectories at each time step. We explore the dependence of the system behavior on different model parameters that capture key recovery strategies. The success probability of the recovery of a failed node needs to be high enough for the network to restore its original functionality. Similarly, the increase in recovery budget parameter also leads to faster and higher recovery levels. However, in most cases, there appears to be upper limits for both parameters, beyond which any further increase could not improve the recovery performance. There is an optimum portion of the loads of the active neighboring nodes that will be carried by the newly recovered node that results in the shortest recovery times or highest recovery levels. Our work sheds light on how to enhance networked systems resiliency by considering the optimization of various model parameters.

Prediction and mitigation of nonlocal cascading failures using graph neural networks

Cascading failures in electrical power grids, comprising nodes and links, propagate nonlocally. After a local disturbance, successive resultant can be distant from the source. Since avalanche failures can propagate unexpectedly, care must be taken when formulating a mitigation strategy. Herein, we propose a strategy for mitigating such cascading failures. First, to characterize the impact of each node on the avalanche dynamics, we propose a novel measure, that of Avalanche Centrality (AC). Then, based on the ACs, nodes potentially needing reinforcement are identified and selected for mitigation. Compared with heuristic measures, AC has proven to be efficient at reducing avalanche size; however, due to nonlocal propagation, calculating ACs can be computationally burdensome. To resolve this problem, we use a graph neural network (GNN). We begin by training a GNN using a large number of small networks; then, once trained, the GNN can predict ACs efficiently in large networks and real-world topological power grids in manageable computational time. Thus, under our strategy, mitigation in large networks is achieved by reinforcing nodes with large ACs. The framework developed in this study can be implemented in other complex processes that require longer computational time to simulate large networks.

A model for cascading failures with the probability of failure described as a logistic function

In most cascading failure models in networks, overloaded nodes are assumed to fail and are removed from the network. However, this is not always the case due to network mitigation measures. Considering the effects of these mitigating measures, we propose a new cascading failure model that describes the probability that an overloaded node fails as a logistic function. By performing numerical simulations of cascading failures on Barabási and Albert (BA) scale-free networks and a real airport network, we compare the results of our model and the established model describing the probability of failure as a linear function. The simulation results show that the difference in the robustness of the two models depends on the initial load distribution and the redistribution of load. We further investigate the conditions of our new model under which the network exhibits the strongest robustness in terms of the load distribution and the network topology. We find the optimal value for the parameter of the load distribution and demonstrate that the robustness of the network improves as the average degree increases. The results regarding the optimal load distribution are verified by theoretical analysis. This work can be used to develop effective mitigation measures and design networks that are robust to cascading failure phenomena.

Graph Metrics for Network Robustness—A Survey

Research on the robustness of networks, and in particular the Internet, has gained critical importance in recent decades because more and more individuals, societies and firms rely on this global network infrastructure for communication, knowledge transfer, business processes and e-commerce. In particular, modeling the structure of the Internet has inspired several novel graph metrics for assessing important topological robustness features of large complex networks. This survey provides a comparative overview of these metrics, presents their strengths and limitations for analyzing the robustness of the Internet topology, and outlines a conceptual tool set in order to facilitate their future adoption by Internet research and practice but also other areas of network science.

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.

Generalized network dismantling

The proper functioning of many sociotechnical systems depends on their level of connectivity. By removing or deactivating a specific set of nodes, a network structure can be dismantled into isolated subcomponents, thereby disrupting the malfunctioning of a system or containing the spread of misinformation or an epidemic. We propose a generalized network-dismantling framework, which can take realistic removal costs into account such as the node price, the protection level, or removal energy. We discuss applications of cost-efficient dismantling strategies to real-world problems such as containing an epidemic or dismantling criminal or corruption networks.

Finding an optimal subset of nodes in a network that is able to efficiently disrupt the functioning of a corrupt or criminal organization or contain an epidemic or the spread of misinformation is a highly relevant problem of network science. In this paper, we address the generalized network-dismantling problem, which aims at finding a set of nodes whose removal from the network results in the fragmentation of the network into subcritical network components at minimal overall cost. Compared with previous formulations, we allow the costs of node removals to take arbitrary nonnegative real values, which may depend on topological properties such as node centrality or on nontopological features such as the price or protection level of a node. Interestingly, we show that nonunit costs imply a significantly different dismantling strategy. To solve this optimization problem, we propose a method which is based on the spectral properties of a node-weighted Laplacian operator and combine it with a fine-tuning mechanism related to the weighted vertex cover problem. The proposed method is applicable to large-scale networks with millions of nodes. It outperforms current state-of-the-art methods and opens more directions for understanding the vulnerability and robustness of complex systems.

Resilience of and recovery strategies for weighted networks

The robustness and resilience of complex networks have been widely studied and discussed in both research and industry because today, the diversity of system components and the complexity of the connection between units are increasingly influencing the reliability of complex systems. Previous studies have focused on node failure in networks, proposing several performance indicators. However, a single performance indicator cannot comprehensively measure all the performance aspects; thus, the selected performance indicators and recovery strategies lack consistency with respect to the targeted complex systems. This paper introduces a novel stress–strength-balanced weighted network model based on two network transmission hypotheses. Then, with respect to different concerns of the complex network, we propose two modified network performance measurement indicators and compare these indicators in terms of their trends in the attack process. In addition, we introduce several network recovery strategies and compare their efficiencies. Our findings are as follows: (1) The evaluation and judgment of the network performance depend on the performance measurement indicators we use. (2) Different recovery strategies exhibit distinct efficiencies in recovering different aspects of network performance, and no strategy exists that can improve all the network performance aspects simultaneously. (3) The timing of the recovery is proved to have a deep influence on the cost and efficiency of network recovery; thus, the optimal recovery strategy for a damaged network varies with the extent of the damage. From the results of the simulation of the attack-recovery process, we conclude that while defining and analyzing complex network models, we should adjust our network topology, weight assignment, and performance indicators in accordance with the focal characteristics of complex systems so that we can use the network model to build robust complex systems and efficient logistics and maintenance strategies.

Network Dynamics of Attention During a Naturalistic Behavioral Paradigm

This study investigates the dynamics of attention during continuous, naturalistic interactions in a video game. Specifically, the effect of repeated distraction on a continuous primary task is related to a functional model of network connectivity. We introduce the Non-linear Attentional Saturation Hypothesis (NASH), which predicts that effective connectivity within attentional networks increases non-linearly with decreasing distraction over time, and exhibits dampening at critical parameter values. Functional magnetic resonance imaging (fMRI) data collected using a naturalistic behavioral paradigm coupled with an interactive video game is used to test the hypothesis. As predicted, connectivity in pre-defined regions corresponding to attentional networks increases as distraction decreases. Moreover, the functional relationship between connectivity and distraction is convex, that is, network connectivity somewhat increases as distraction decreases during the continuous primary task, however, connectivity increases considerably as distraction falls below critical levels. This result characterizes the non-linear pattern of connectivity within attentional networks, particularly with respect to their dynamics during behavior. These results are also summarized in the form of a network structure analysis, which underscores the role of various nodes in regulating the global network state. In conclusion, we situate the implications of this research in the context of cognitive complexity and an emerging theory of flow during media exposure.

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.

Overload-based cascades on multiplex networks and effects of inter-similarity

Although cascading failures caused by overload on interdependent/interconnected networks have been studied in the recent years, the effect of overlapping links (inter-similarity) on robustness to such cascades in coupled networks is not well understood. This is an important issue since shared links exist in many real-world coupled networks. In this paper, we propose a new model for load-based cascading failures in multiplex networks. We leverage it to compare different network structures, coupling schemes, and overload rules. More importantly, we systematically investigate the impact of inter-similarity on the robustness of the whole system under an initial intentional attack. Surprisingly, we find that inter-similarity can have a negative impact on robustness to overload cascades. To the best of our knowledge, we are the first to report the competition between the positive and the negative impacts of overlapping links on the robustness of coupled networks. These results provide useful suggestions for designing robust coupled traffic systems.

Mitigate Cascading Failures on Networks using a Memetic Algorithm

Research concerning cascading failures in complex networks has become a hot topic. However, most of the existing studies have focused on modelling the cascading phenomenon on networks and analysing network robustness from a theoretical point of view, which considers only the damage incurred by the failure of one or several nodes. However, such a theoretical approach may not be useful in practical situation. Thus, we first design a much more practical measure to evaluate the robustness of networks against cascading failures, termed Rcf. Then, adopting Rcf as the objective function, we propose a new memetic algorithm (MA) named MA-Rcf to enhance network the robustness against cascading failures. Moreover, we design a new local search operator that considers the characteristics of cascading failures and operates by connecting nodes with a high probability of having similar loads. In experiments, both synthetic scale-free networks and real-world networks are used to test the efficiency and effectiveness of the MA-Rcf. We systematically investigate the effects of parameters on the performance of the MA-Rcf and validate the performance of the newly designed local search operator. The results show that the local search operator is effective, that MA-Rcf can enhance network robustness against cascading failures efficiently, and that it outperforms existing algorithms.

Advanced Algorithms for Local Routing Strategy on Complex Networks

Despite the significant improvement on network performance provided by global routing strategies, their applications are still limited to small-scale networks, due to the need for acquiring global information of the network which grows and changes rapidly with time. Local routing strategies, however, need much less local information, though their transmission efficiency and network capacity are much lower than that of global routing strategies. In view of this, three algorithms are proposed and a thorough investigation is conducted in this paper. These algorithms include a node duplication avoidance algorithm, a next-nearest-neighbor algorithm and a restrictive queue length algorithm. After applying them to typical local routing strategies, the critical generation rate of information packets R_c increases by over ten-fold and the average transmission time 〈T〉 decreases by 70–90 percent, both of which are key physical quantities to assess the efficiency of routing strategies on complex networks. More importantly, in comparison with global routing strategies, the improved local routing strategies can yield better network performance under certain circumstances. This is a revolutionary leap for communication networks, because local routing strategy enjoys great superiority over global routing strategy not only in terms of the reduction of computational expense, but also in terms of the flexibility of implementation, especially for large-scale networks.

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.

Nonlocal effects and countermeasures in cascading failures

We study the propagation of cascading failures in complex supply networks with a focus on nonlocal effects occurring far away from the initial failure. It is shown that a high clustering and a small average path length of a network generally suppress nonlocal overloads. These properties are typical for many real-world networks, often called small-world networks, such that cascades propagate mostly locally in these networks. Furthermore, we analyze the spatial aspects of countermeasures based on the intentional removal of additional edges. Nonlocal actions are generally required in networks that have a low redundancy and are thus especially vulnerable to cascades.

Controlling extreme events on complex networks

Extreme events, a type of collective behavior in complex networked dynamical systems, often can have catastrophic consequences. To develop effective strategies to control extreme events is of fundamental importance and practical interest. Utilizing transportation dynamics on complex networks as a prototypical setting, we find that making the network “mobile” can effectively suppress extreme events. A striking, resonance-like phenomenon is uncovered, where an optimal degree of mobility exists for which the probability of extreme events is minimized. We derive an analytic theory to understand the mechanism of control at a detailed and quantitative level, and validate the theory numerically. Implications of our finding to current areas such as cybersecurity are discussed.

Dynamic source routing strategy for two-level flows on scale-free networks

Packets transmitting in real communication networks such as the Internet can be classified as time-sensitive or time-insensitive. To better support the real-time and time-insensitive applications, we propose a two-level flow traffic model in which packets are labeled as level-1 or level-2, and those with level-1 have higher priority to be transmitted. In order to enhance the traffic capacity of the two-level flow traffic model, we expand the global dynamic routing strategy and propose a new dynamic source routing which supports no routing-flaps, high traffic capacity, and diverse traffic flows. As shown in this paper, the proposed dynamic source routing can significantly enhance the traffic capacity and quality of time-sensitive applications compared with the global shortest path routing strategy.

Manipulation of extreme events on scale-free networks

Extreme events taking place on networks are not uncommon. We show that it is possible to manipulate the extreme event occurrence probabilities and distribution over the nodes of a scale-free network by tuning the nodal capacity. This can be used to reduce the number of extreme event occurrences. However, monotonic nodal capacity enhancements, beyond a point, do not lead to any substantial reduction in the number of extreme events. We point out the practical implication of this result for network design in the context of reducing extreme event occurrences.

Structural robustness of scale-free networks against overload failures

We study the structural robustness of scale-free networks against overload failures induced by loads exceeding the node capacity, based on analytical and numerical approaches to the percolation problem in which a fixed number of nodes are removed according to the overload probability. Modeling fluctuating loads by random walkers in a network, we find that the degree dependence of the overload probability drastically changes with respect to the total load. We also elucidate that there exist two types of structural robustness of networks against overload failures. One is measured by the critical total load W(c) and the other is by the critical node removal fraction f(c). Enhancing the scale-free property, networks become fragile in both senses of W(c) and f(c). By contrast, increasing the node tolerance, scale-free networks become robust in the sense of the critical total load, while they come to be fragile in the sense of the critical node removal fraction. Furthermore, we show that these trends are not affected by degree-degree correlations, although assortative mixing makes networks robust in both senses of W(c) and f(c).

Universality of flux-fluctuation law in complex dynamical systems

Recent work has revealed a law governing flux fluctuation and the average flux in complex dynamical systems. We establish the universality of this flux-fluctuation law through the following steps: (i) We derive the law in a more general setting, showing that it depends on a single parameter characterizing the external driving; (ii) we conduct extensive numerical computations using distinct external driving, different network topologies, and multiple traffic routing strategies; and (iii) we analyze data from an actual vehicle traffic system in a major city in China to lend more credence to the universality of the flux-fluctuation law. Additional factors considered include flux fluctuation on links, window size effect, and hidden topological structures such as nodal degree correlation. Besides its fundamental importance in complex systems, the flux-fluctuation law can be used to infer certain intrinsic property of the system for potential applications such as control of complex systems for improved performance.

Analysis of cascading failure in gene networks

It is an important subject to research the functional mechanism of cancer-related genes make in formation and development of cancers. The modern methodology of data analysis plays a very important role for deducing the relationship between cancers and cancer-related genes and analyzing functional mechanism of genome. In this research, we construct mutual information networks using gene expression profiles of glioblast and renal in normal condition and cancer conditions. We investigate the relationship between structure and robustness in gene networks of the two tissues using a cascading failure model based on betweenness centrality. Define some important parameters such as the percentage of failure nodes of the network, the average size-ratio of cascading failure, and the cumulative probability of size-ratio of cascading failure to measure the robustness of the networks. By comparing control group and experiment groups, we find that the networks of experiment groups are more robust than that of control group. The gene that can cause large scale failure is called structural key gene. Some of them have been confirmed to be closely related to the formation and development of glioma and renal cancer respectively. Most of them are predicted to play important roles during the formation of glioma and renal cancer, maybe the oncogenes, suppressor genes, and other cancer candidate genes in the glioma and renal cancer cells. However, these studies provide little information about the detailed roles of identified cancer genes.

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.

Network growth under the constraint of synchronization stability

While it is well recognized that realistic networks are typically growing with time, the dynamical features of their growing processes remain to be explored. In the present paper, incorporating the requirement of synchronization stability into the conventional models of network growth, we will investigate how the growing process of a complex network is influenced by, and also will influence, the network collective dynamics. Our study shows that, constrained by the synchronization stability, the network will be growing in a selective and dynamical fashion. In particular, we find that the chance for a new node to be accepted by the growing network could have a large variation, i.e., it follows roughly a power-law distribution. Furthermore, we find that, with the dynamical growth, the network is always developed into structures of clear scale-free features, despite the form of the link attachment (preferential or random). The dynamical properties of network growth are studied using the method of eigenvalue analysis, and they are verified by direct simulations of coupled chaotic oscillators. Our study implies that, driven by the network collective dynamics, network growth could also be highly dynamical.

Cascades on a class of clustered random networks

We present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of random networks with arbitrary degree distribution and nonzero clustering introduced previously in [M. E. J. Newman, Phys. Rev. Lett. 103, 058701 (2009)]. A condition for the existence of global cascades is derived as well as a general criterion that determines whether increasing the level of clustering will increase, or decrease, the expected cascade size. Applications, examples of which are provided, include site percolation, bond percolation, and Watts' threshold model; in all cases analytical results give excellent agreement with numerical simulations.

The role of nonlinearity in computing graph-theoretical properties of resting-state functional magnetic resonance imaging brain networks

In recent years, there has been an increasing interest in the study of large-scale brain activity interaction structure from the perspective of complex networks, based on functional magnetic resonance imaging (fMRI) measurements. To assess the strength of interaction (functional connectivity, FC) between two brain regions, the linear (Pearson) correlation coefficient of the respective time series is most commonly used. Since a potential use of nonlinear FC measures has recently been discussed in this and other fields, the question arises whether particular nonlinear FC measures would be more informative for the graph analysis than linear ones. We present a comparison of network analysis results obtained from the brain connectivity graphs capturing either full (both linear and nonlinear) or only linear connectivity using 24 sessions of human resting-state fMRI. For each session, a matrix of full connectivity between 90 anatomical parcel time series is computed using mutual information. For comparison, connectivity matrices obtained for multivariate linear Gaussian surrogate data that preserve the correlations, but remove any nonlinearity are generated. Binarizing these matrices using multiple thresholds, we generate graphs corresponding to linear and full nonlinear interaction structures. The effect of neglecting nonlinearity is then assessed by comparing the values of a range of graph-theoretical measures evaluated for both types of graphs. Statistical comparisons suggest a potential effect of nonlinearity on the local measures-clustering coefficient and betweenness centrality. Nevertheless, subsequent quantitative comparison shows that the nonlinearity effect is practically negligible when compared to the intersubject variability of the graph measures. Further, on the group-average graph level, the nonlinearity effect is unnoticeable.

Global dynamic routing for scale-free networks

Traffic is essential for many dynamic processes on networks. The efficient routing strategy [G. Yan, T. Zhou, B. Hu, Z. Q. Fu, and B. H. Wang, Phys. Rev. E 73, 046108 (2006)] can reach a very high capacity of more than ten times of that with shortest path strategy. In this paper, we propose a global dynamic routing strategy for network systems based on the information of the queue length of nodes. Under this routing strategy, the traffic capacity is further improved. With time delay of updating node queue lengths and the corresponding paths, the system capacity remains constant, while the travel time for packets increases.

Pheromone routing protocol on a scale-free network

This paper proposes a routing strategy for network systems based on the local information of "pheromone." The overall traffic capacity of a network system can be evaluated by the critical packet generating rate R(c). Under this critical generating rate, the total packet number in the system first increases and then decreases to reach a balance state. The system behaves differently from that with a local routing strategy based on the node degree or shortest path routing strategy. Moreover, the pheromone routing strategy performs much better than the local routing strategy, which is demonstrated by a larger value of the critical generating rate. This protocol can be an alternation for superlarge networks, in which the global topology may not be available.

Abrupt transition to complete congestion on complex networks and control

Previous works on traffic-flow dynamics on complex networks have mostly focused on continuous phase transition from a free-flow state to a locally congested state as a parameter, such as the packet-generating rate, is increased through a critical value. Above the transition point congestion occurs on a small subset of nodes. Utilizing a conventional traffic-flow model based on the packet birth-death process and more importantly, taking into account the fact that in realistic networks nodes have only finite buffers, we find an abrupt transition from free flow to complete congestion. Slightly below the transition point, the network can support the maximum amount of traffic for some optimal value of the routing parameter. We develop a mean-field theory to explain the surprising transition phenomenon and provide numerical support. Furthermore, we propose a control strategy based on the idea of random packet dropping to prevent/break complete congestion. Our finding provides insights into realistic communication networks where complete congestion can occur directly from a free-flow state without any apparent precursor, and our control strategy can be effective to restore traffic flow once complete congestion has occurred.

(Psi,p,q)-vulnerabilities: a unified approach to network robustness

We define a new general framework, the family of (psi,p,q)-vulnerabilities, as a tool that produces many new vulnerability functions that measure the capacity of a network to maintain its functional performance under random damages or malicious attacks. This new framework comprises most of the vulnerability definitions appearing in the literature and allows to calculate some relationships between the different (psi,p,q)-vulnerabilities in terms of their function psi or their parameters p, q that improve several known results for the vulnerability functions. Some graphics of simulations are provided in order to show the sharpness of these relationships.

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

This paper is motivated by the following two related problems in complex networks: (i) control of cascading failures and (ii) mitigation of traffic congestion. Both problems are of significant recent interest as they address, respectively, the security of and efficient information transmission on complex networks. Taking into account typical features of load distribution and weights in real-world networks, we have discovered an optimal solution to both problems. In particular, we shall provide numerical evidence and theoretical analysis that, by choosing a proper weighting parameter, a maximum level of robustness against cascades and traffic congestion can be achieved, which practically rids the network of occurrences of the catastrophic dynamics.

Cascades on correlated and modular random networks

An analytical approach to determining the mean avalanche size in a broad class of dynamical models on random networks is introduced. Previous results on percolation transitions and epidemic sizes are shown to be special cases of the method. The time-dependence of cascades and extensions to networks with community structure or degree-degree correlations are discussed. Analytical results for the rate of spread of innovations in a modular network and for the size of k cores in networks with degree-degree correlations are confirmed with numerical simulations.

Universal robustness characteristic of weighted networks against cascading failure

We investigate the cascading failure on weighted complex networks by adopting a local weighted flow redistribution rule, where the weight of an edge is (k(i)k(j))theta with k(i) and k(j) being the degrees of the nodes connected by the edge. Assume that a failed edge leads only to a redistribution of the flow passing through it to its neighboring edges. We found that the weighted complex network reaches the strongest robustness level when the weight parameter theta=1, where the robustness is quantified by a transition from normal state to collapse. We determined that this is a universal phenomenon for all typical network models, such as small-world and scale-free networks. We then confirm by theoretical predictions this universal robustness characteristic observed in simulations. We furthermore explore the statistical characteristics of the avalanche size of a network, thus obtaining a power-law avalanche size distribution together with a tunable exponent by varying theta. Our findings have great generality for characterizing cascading-failure-induced disasters in nature.

Complex network synchronizability: analysis and control

In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution, and node betweenness centrality, but have very different synchronizabilities. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.

Error and attack tolerance of layered complex networks

Many complex systems may be described by not one but a number of complex networks mapped on each other in a multi-layer structure. Because of the interactions and dependencies between these layers, the state of a single layer does not necessarily reflect well the state of the entire system. In this paper we study the robustness of five examples of two-layer complex systems: three real-life data sets in the fields of communication (the Internet), transportation (the European railway system), and biology (the human brain), and two models based on random graphs. In order to cover the whole range of features specific to these systems, we focus on two extreme policies of system's response to failures, no rerouting and full rerouting. Our main finding is that multi-layer systems are much more vulnerable to errors and intentional attacks than they appear from a single layer perspective.

Seed size strongly affects cascades on random networks

The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree z of the network is known to exhibit several transitions-these are typically continuous at low z and discontinuous at high z; here it is demonstrated that the low- z transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero-temperature random-field Ising model on random graphs are discussed.

Oscillations of complex networks

A complex network processing information or physical flows is usually characterized by a number of macroscopic quantities such as the diameter and the betweenness centrality. An issue of significant theoretical and practical interest is how such quantities respond to sudden changes caused by attacks or disturbances in recoverable networks, i.e., functions of the affected nodes are only temporarily disabled or partially limited. By introducing a model to address this issue, we find that, for a finite-capacity network, perturbations can cause the network to oscillate persistently in the sense that the characterizing quantities vary periodically or randomly with time. We provide a theoretical estimate of the critical capacity-parameter value for the onset of the network oscillation. The finding is expected to have broad implications as it suggests that complex networks may be structurally highly dynamic.

Extraction and analysis of traffic and topologies of transportation networks

The knowledge of real-life traffic patterns is crucial for a good understanding and analysis of transportation systems. These data are quite rare. In this paper we propose an algorithm for extracting both the real physical topology and the network of traffic flows from timetables of public mass transportation systems. We apply this algorithm to timetables of three large transportation networks. This enables us to make a systematic comparison between three different approaches to construct a graph representation of a transportation network; the resulting graphs are fundamentally different. We also find that the real-life traffic pattern is very heterogenous, in both space and traffic flow intensities, which makes it very difficult to approximate the node load with a number of topological estimators.

Integrating local static and dynamic information for routing traffic

The efficiency of traffic routing on complex networks can be reflected by two key measurements, i.e., the network capacity and the average travel time of data packets. In this paper we propose a mixing routing strategy by integrating local static and dynamic information for enhancing the efficiency of traffic on scale-free networks. The strategy is governed by a single parameter. Simulation results show that maximizing the network capacity and reducing the packet travel time can generate an optimal parameter value. Compared with the strategy of adopting exclusive local static information, the new strategy shows its advantages in improving the efficiency of the system. The detailed analysis of the mixing strategy is provided for explaining its effects on traffic routing. The work indicates that effectively utilizing the larger degree nodes plays a key role in scale-free traffic systems.

Percolation and blind spots in complex networks

Recent works on network security have focused on whether a complex network can maintain its integrability under attack or random node failures. In applications of increasing importance such as sensor networks, a somewhat different problem, namely, the occurrence of isolated nodes (or blind spots), is of great interest. We show that, for networks with a stronger ability to form global spanning clusters, it is relatively more difficult to eliminate blind spots, and vice versa. We use the framework of percolation to investigate this phenomenon. Our analysis also yields a formula for the average number of blind spots, which provide an explanation for several numerical findings.

Layered complex networks

Many complex networks are only a part of larger systems, where a number of coexisting topologies interact and depend on each other. We introduce a layered model to facilitate the description and analysis of such systems. As an example of its application, we study the load distribution in three transportation systems, where the lower layer is the physical infrastructure and the upper layer represents the traffic flows. This layered view allows us to capture the fundamental differences between the real load and commonly used load estimators, which explains why these estimators fail to approximate the real load.

Efficient routing on complex networks

We propose a routing strategy to improve the transportation efficiency on complex networks. Instead of using the routing strategy for shortest path, we give a generalized routing algorithm to find the so-called efficient path, which considers the possible congestion in the nodes along actual paths. Since the nodes with the largest degree are very susceptible to traffic congestion, an effective way to improve traffic and control congestion, as our strategy, can be redistributing traffic load in central nodes to other noncentral nodes. Simulation results indicate that the network capability in processing traffic is improved more than 10 times by optimizing the efficient path, which is in good agreement with the analysis.

Geographical effects on cascading breakdowns of scale-free networks

Cascading breakdowns of real networks have resulted in severe accidents in recent years. In this paper, we study the effects of geographical structure on the cascading phenomena of load-carrying scale-free networks. Our essential finding is that when networks are more geographically constrained, i.e., more locally interconnected, they tend to have larger cascading breakdowns. Explanations are provided in terms of the effects of cycles and the distributions of betweenness over degrees.

Traffic dynamics based on local routing protocol on a scale-free network

We propose a packet routing strategy with a tunable parameter based on the local structural information of a scale-free network. As free traffic flow on the communication networks is key to their normal and efficient functioning, we focus on the network capacity that can be measured by the critical point of phase transition from free flow to congestion. Simulations show that the maximal capacity corresponds to alpha= -1 in the case of identical nodes' delivering ability. To explain this, we investigate the number of packets of each node depending on its degree in the free flow state and observe the power law behavior. Other dynamic properties including average packets traveling time and traffic load are also studied. Inspiringly, our results indicate that some fundamental relationships exist between the dynamics of synchronization and traffic on the scale-free networks.

Mutual selection model for weighted networks

For most networks, the connection between two nodes is the result of their mutual affinity and attachment. In this paper, we propose a mutual selection model to characterize the weighted networks. By introducing a general mechanism of mutual selection, the model can produce power-law distributions of degree, weight, and strength, as confirmed in many real networks. Moreover, we also obtained the nontrivial clustering coefficient C, degree assortativity coefficient r, and degree-strength correlation depending on a single parameter m. These results are supported by present empirical evidence. Studying the degree-dependent average clustering coefficient C(k) and the degree-dependent average nearest neighbors' degree k(nn)(k) also provide us with a better description of the hierarchies and organizational architecture of weighted networks.

Tolerance of scale-free networks against attack-induced cascades

Scale-free networks can be disintegrated by attack on a single or a very few nodes through the process of cascading failures. By utilizing a prototype cascading model, we previously determined the critical value of the capacity parameter below which the network can become disintegrated due to attack on a single node. A fundamental question in network security, which has not been addressed previously but may be more important and of wider interest, is how to design networks of finite capacity that are safe against cascading breakdown. Here we derive an upper bound for the capacity parameter, above which the network is immune to cascading breakdown. Our theory also yields estimates for the maximally achievable network integrity via controlled removal of a small set of low-degree nodes. The theoretical results are confirmed numerically.

Robustness of the avalanche dynamics in data-packet transport on scale-free networks

We study the avalanche dynamics in the data-packet transport on scale-free networks through a simple model. In the model, each vertex is assigned a capacity proportional to the load with the proportionality constant 1+a . When the system is perturbed by a single vertex removal, the load of each vertex is redistributed, followed by subsequent failures of overloaded vertices. The avalanche size depends on the parameter a as well as which vertex triggers it. We find that there exists a critical value a(c) at which the avalanche size distribution follows a power law. The critical exponent associated with it appears to be robust as long as the degree exponent is between 2 and 3 and is close in value to that of the distribution of the diameter changes by single vertex removal.

Maximal planar networks with large clustering coefficient and power-law degree distribution

In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called random Apollonian networks (RANs) as they can be considered as a variation of Apollonian networks. We obtain the analytic results of power-law exponent gamma=3 and clustering coefficient C= (46/3)-36 ln 3/2 approximately 0.74, which agree with the simulation results very well. We prove that the increasing tendency of average distance of RANs is a little slower than the logarithm of the number of nodes in RANs. Since most real-life networks are both scale-free and small-world networks, RANs may perform well in mimicking the reality. The RANs possess hierarchical structure as C(k) approximately k(-1) that are in accord with the observations of many real-life networks. In addition, we prove that RANs are maximal planar networks, which are of particular practicability for layout of printed circuits and so on. The percolation and epidemic spreading process are also studied and the comparisons between RANs and Barabási-Albert (BA) as well as Newman-Watts (NW) networks are shown. We find that, when the network order N (the total number of nodes) is relatively small (as N approximately 10(4)), the performance of RANs under intentional attack is not sensitive to N , while that of BA networks is much affected by N. And the diseases spread slower in RANs than BA networks in the early stage of the susceptible-infected process, indicating that the large clustering coefficient may slow the spreading velocity, especially in the outbreaks.

Universality class of the fiber bundle model on complex networks

We investigate the failure characteristics of complex networks within the framework of the fiber bundle model subject to the local load sharing rule in which the load of the broken fiber is transferred only to its neighbor fibers. Although the load sharing is strictly local, it is found that the critical behavior belongs to the universality class of global load sharing where the load is transferred equally to all fibers in the system. From the numerical simulations and the analytical approach applied to the microscopic behavior, it is revealed that the emergence of a single dominant hub cluster of broken fibers causes the global load sharing effect in the failure process.

Accurately modeling the internet topology

Based on measurements of the internet topology data, we found that there are two mechanisms which are necessary for the correct modeling of the internet topology at the autonomous systems (AS) level: the interactive growth of new nodes and new internal links, and a nonlinear preferential attachment, where the preference probability is described by a positive-feedback mechanism. Based on the above mechanisms, we introduce the positive-feedback preference (PFP) model which accurately reproduces many topological properties of the AS-level internet, including degree distribution, rich-club connectivity, the maximum degree, shortest path length, short cycles, disassortative mixing, and betweenness centrality. The PFP model is a phenomenological model which provides an insight into the evolutionary dynamics of real complex networks.

Cascading failures in coupled map lattices

Large cascades triggered by initial shocks are common in complex networks. Coupled map lattices have been widely used over the past decades as dynamical models of complex systems. Here we investigate cascading failures in coupled map lattices with different topologies. We find that cascading failures are much easier to occur in small-world and scale-free coupled map lattices than in globally coupled map lattices.

Attack vulnerability of scale-free networks due to cascading breakdown

The possibility that a complex network can be brought down by attack on a single or a very few nodes through the process of cascading failures is of significant concern. Here we investigate a recent model for cascading failures in complex networks and uncover a phase-transition phenomenon in terms of the key parameter characterizing the node capacity. For parameter value below the phase-transition point, cascading failures can cause the network to disintegrate almost entirely. We obtain a theoretical estimate for the phase-transition point and provide numerical support.