Response of scale-free networks with community structure to external stimuli

The response of scale-free networks with community structure to external stimuli is studied. By disturbing some nodes with different strategies, it is shown that the robustness of this kind of network can be enhanced due to the existence of communities in the networks. Some of the response patterns are found to coincide with topological communities. We show that such phenomena also occur in the cat brain network which is an example of a scale-free like network with community structure. Our results provide insights into the relationship between network topology and the functional organization in complex networks from another viewpoint.     

A Robustness Model of Complex Networks with Tunable Attack Information Parameter

We introduce a novel model for robustness of complex with a tunable attack information parameter. The random failure and intentional attack known are the two extreme cases of our model. Based on the model, we study the robustness of complex networks under random information and preferential information, respectively. Using the generating function method, we derive the exact value of the critical removal fraction of nodes for the disintegration of networks and the size of the giant component. We show that hiding just a small fraction of nodes randomly can prevent a scale-free network from collapsing and detecting just a small fraction of nodes preferentially can destroy a scale-free network.     

Robustness of Complex Networks under Attack and Repair

To study the robustness of complex networks under attack and repair, we introduce a repair model of complex networks. Based on the model, we introduce two new quantities, i.e. attack fraction fa and the maximum degree of the nodes that have never been attacked ~Ka, to study analytically the critical attack fraction and the relative size of the giant component of complex networks under attack and repair, using the method of generating function. We show analytically and numerically that the repair strategy significantly enhances the robustness of the scale-free network and the effect of robustness improvement is better for the scale-free networks with a smaller degree exponent. We discuss the application of our theory in relation to the
understanding of robustness of complex networks with reparability.     

Complex Networks: from Graph Theory to Biology

The aim of this text is to show the central role played by networks in complex system science. A remarkable feature of network studies is to lie at the crossroads of different disciplines, from mathematics (graph theory, combinatorics, probability theory) to physics (statistical physics of networks) to computer science (network generating algorithms, combinatorial optimization) to biological issues (regulatory networks). New paradigms recently appeared, like that of ‘scale-free networks’ providing an alternative to the random graph model introduced long ago by Erdös and Renyi. With the notion of statistical ensemble and methods originally introduced for percolation networks, statistical physics is of high relevance to get a deep account of topological and statistical properties of a network. Then their consequences on the dynamics taking place in the network should be investigated. Impact of network theory is huge in all natural sciences, especially in biology with gene networks, metabolic networks, neural networks or food webs. I illustrate this brief overview with a recent work on the influence of network topology on the dynamics of coupled excitable units, and the insights it provides about network emerging features, robustness of network behaviors, and the notion of static or dynamic motif.     

Synchronizability of dynamical scale-free networks subject to random errors

In this paper the robustness of network synchronizability against random deletion of nodes, i.e. errors, in dynamical scale-free networks was studied. To this end, two measures of network synchronizability, namely, the eigenratio of the Laplacian and the order parameter quantifying the degree of phase synchrony were adopted, and the synchronizability robustness on preferential attachment scale-free graphs was investigated. The findings revealed that as the network size decreases, the robustness of its synchronizability against random removal of nodes declines, i.e. the more the number of randomly removed nodes from the network, the worse its synchronizability. We also showed that this dependence of the synchronizability on the network size is different with that in the growing scale-free networks. The profile of a number of network properties such as clustering coefficient, efficiency, assortativity, and eccentricity, as a function of the network size was investigated in these two cases, growing scale-free networks and those with randomly removed nodes. The results showed that these processes are also different in terms of these metrics.     

Effect of Eliminating Edges on Robustness of Scale-Free Networks under Intentional Attack

We study the robustness of complex networks under edge elimination. We propose three different edge elimination strategies and investigate their effects on the robustness of scale-free networks under intentional attack. We show that deleting a proper fraction of edges connecting hub nodes and hub nodes can enhance the robustness of scale-free networks under intentional attack.     

小世界网络与无标度网络的社区结构研究

模块性(modularity)是度量网络社区结构(community structure)的主要参数.探讨了Watts和Strogatz的小世界网络(简称W-S模型)以及Barabàsi 等的B-A无标度网络(简称B-A模型)两类典型复杂网络模块性特点.结果显示，网络模块性受到网络连接稀疏的影响，W-S模型具有显著的社区结构，而B-A模型的社区结构特征不明显.因此，应用中应该分别讨论网络的小世界现象和无标度特性.社区结构不同于小世界现象和无标度特性，并可以利用模块性区别网络类型，因此网络复杂性指标应该包括 关键词： 模块性 社区结构 小世界网络 无标度网络     

Cascading Failures of Complex Networks Based on Two-Step Degree

We propose a new concept, two-step degree. Defining it as the capacity of a node of complex networks, we establish a novel capacity-load model of cascading failures of complex networks where the capacity of nodes decreases during the process of cascading failures. For scale-free networks, we find that the average two-step degree increases with the increase of the heterogeneity of the degree distribution, showing that the average two- step degree can be used for measuring the heterogeneity of the degree distribution of complex networks. In addition, under the condition that the average degree of a node is given, we can design a scale-free network with the optimal robustness to random failures by maximizing the average two-step degree.     

Model for cascading failures with adaptive defense in complex networks

<正>This paper investigates cascading failures in networks by considering interplay between the flow dynamic and the network topology,where the fluxes exchanged between a pair of nodes can be adaptively adjusted depending on the changes of the shortest path lengths between them.The simulations on both an artificially created scale-free network and the real network structure of the power grid reveal that the adaptive adjustment of the fluxes can drastically enhance the robustness of complex networks against cascading failures.Particularly,there exists an optimal region where the propagation of the cascade is significantly suppressed and the fluxes supported by the network are maximal. With this understanding,a costless strategy of defense for preventing cascade breakdown is proposed.It is shown to be more effective for suppressing the propagation of the cascade than the recent proposed strategy of defense based on the intentional removal of nodes.     

A robust matching model of capacity to defense cascading failure on complex networks

How to control the cascading failure has become a hot topic in recent years. In this paper, we propose a new matching model of capacity by developing a profit function to defense cascading failures on artificially created scale-free networks and the real network structure of the North American power grid. Results show that our matching model can enhance the network robustness efficiently, which is particularly important for the design of networks to deduce the damage triggered by the cascading failures.     

Optimization in gradient networks

Gradient networks can be used to model the dominant structure of complex networks. Previous work has focused on random gradient networks. Here we study gradient networks that minimize jamming on substrate networks with scale-free and Erdos-Renyi structure. We introduce structural correlations and strongly reduce congestion occurring on the network by using a Monte Carlo optimization scheme. This optimization alters the degree distribution and other structural properties of the resulting gradient networks. These results are expected to be relevant for transport and other dynamical processes in real network systems.     

Efficiency and robustness in ant networks of galleries

Recent theoretical and empirical studies have focused on the topology of large networks of communication/interactions in biological, social and technological systems. Most of them have been studied in the scope of the small-world and scale-free networks theory. Here we analyze the characteristics of ant networks of galleries produced in a 2-D experimental setup. These networks are neither small-worlds nor scale-free networks and belong to a particular class of network, i.e. embedded planar graphs emerging from a distributed growth mechanism. We compare the networks of galleries with both minimal spanning trees and greedy triangulations. We show that the networks of galleries have a path system efficiency and robustness to disconnections closer to the one observed in triangulated networks though their cost is closer to the one of a tree. These networks may have been prevented to evolve toward the classes of small-world and scale-free networks because of the strong spatial constraints under which they grow, but they may share with many real networks a similar trend to result from a balance of constraints leading them to achieve both path system efficiency and robustness at low cost.Received: 16 July 2004, Published online: 26 November 2004PACS: 89.75.Fb Structures and organization in complex systems - 89.75.Hc Networks and genealogical trees - 87.23.Ge Dynamics of social systems     

Enhancement of scale-free network attack tolerance

Despite the large size of most communication and transportation systems,there are short paths between nodes in these networks which guarantee the efficient information,data and passenger delivery;furthermore these networks have a surprising tolerance under random errors thanks to their inherent scale-free topology.However,their scale-free topology also makes them fragile under intentional attacks,leaving us a challenge on how to improve the network robustness against intentional attacks without losing their strong tolerance under random errors and high message and passenger delivering capacity.Here we propose two methods (SL method and SH method) to enhance scale-free network’s tolerance under attack in different conditions.     

Optimization of robustness and connectivity in complex networks

Scale-free networks rely on a relatively small number of highly connected nodes to achieve a high degree of interconnectivity and robustness to random failure, but suffer from a high sensitivity to directed attack. In this paper we describe a parameterized family of networks and analyze their connectivity and sensitivity, identifying a network that has an interconnectedness closer to that of a scale-free network, a robustness to attack closer to that of an exponential network, and a resistance to failure better than that of either of those networks.     

Mitigation strategies on scale-free networks against cascading failures

According to the dynamic characteristics of the cascading propagation, we introduce a mitigation mechanism and propose four mitigation methods on four types of nodes. By the normalized average avalanche size and a new measure, we demonstrate the efficiencies of the mitigation strategies on enhancing the robustness of scale-free networks against cascading failures and give the order of the effectiveness of the mitigation strategies. Surprisingly, we find that only adopting once mitigation mechanism on a small part of the overload nodes can dramatically improve the robustness of scale-free networks. In addition, we also show by numerical simulations that the optimal mitigation method strongly depends on the total capacities of all nodes in a network and the distribution of the load in the cascading model. Therefore, according to the protection strength for scale-free networks, by the distribution of the load and the protection price of networks, we can reasonably select how many nodes and which mitigation method to efficiently protect scale-free networks at the lower price. These findings may be very useful for avoiding various cascading-failure-induced disasters in the real world and for leading to insights into the mitigation of cascading failures.     

有倾向性重连产生的反匹配网络

在具有网络结构的系统中度关联属性对于动力学行为具有重要的影响, 所以产生适当度关联网络的方法对于大量网络系统的研究具有重要的作用. 尽管产生正匹配网络的方法已经得到很好的验证, 但是产生反匹配网络的方法还没有被系统的讨论过. 重新连接网络中的边是产生度关联网络的一个常用方法. 这里我们研究使用重连方法产生反匹配无标度网络的有效性. 我们的研究表明, 有倾向的重连可以增强网络的反匹配属性. 但是有倾向重连不能使皮尔森度相关系数下降到-1, 而是存在一个依赖于网络参数的最小值. 我们研究了网络的主要参数对于网络度相关系数的影响, 包括网络尺寸, 网络的连接密度和网络节点的度差异程度. 研究表明在网络尺寸大的情况下和节点度差异性强的情况下, 重连的效果较差. 我们研究了真实Internet网络, 发现模型产生的网络经过重连不能达到真实网络的度关联系数.     

Impact of core-periphery structure on cascading failures in interdependent scale-free networks

Core-periphery structure is a typical meso-scale structure in networks. Previous studies on core-periphery structure mainly focus on the improvement of detection methods, while the research on the impact of core-periphery structure on cascading failures in interdependent networks is still missing. Therefore, we investigate the cascading failures of interdependent scale-free networks with different core-periphery structures and coupling preferences in the paper. First, we introduce an evaluation index to calculate the goodness of core-periphery structure. Second, we propose a new scale-free network evolution model, which can generate tunable core-periphery structures, and its degree distribution is analyzed mathematically. Finally, based on a degree-load-based cascading failure model, we mainly investigate the impact of goodness of core-periphery structure on cascading failures in both symmetrical and asymmetrical interdependent networks. Through numerical simulations, we find that with the same average degree, the networks with weak core-periphery structure will be more robust, while the initial load on node will influence the improvement of robustness. In addition, we also find that the inter-similarity coupling performs better than random coupling. These findings may be helpful for building resilient interdependent networks.     

Traffic dynamics on layered complex networks

To minimize traffic congestion, understanding how traffic dynamics depend on network structure is necessary. Many real-world complex systems can be described as multilayer structures. In this paper, we introduce the idea of layers to establish a traffic model of two-layer complex networks. By comparing different two-layer complex networks based on random and scale-free networks, we find that the physical layer is much more important to the network capacity of two-layer complex networks than the logical layer. Two-layer complex networks with a homogeneous physical topology are found to be more tolerant to congestion. Moreover, simulation results show that the heterogeneity of logical and physical topologies makes the packet-delivery process of two-layer networks more efficient in the free-flow state, without the occurrence of traffic congestion.     

Error and attack tolerance of evolving networks with local preferential attachment

Networks generated by local-world evolving network model display a transition from exponential network to power-law network with respect to connectivity distribution. We investigate statistical properties of the evolving networks and the responses of these networks under random errors and intentional attacks. It has been found that local world size M has great effect on the network's heterogeneity, thus leading to transitional behaviors in network's robustness against errors and attacks. Numerical results show that networks constructed with local preferential attachment mechanism can maintain the robustness of scale-free networks under random errors and concurrently improve reliance against targeted attacks on highly connected nodes.     

Epidemic spreading on networks with vaccination

In this paper, a new susceptible-infected-susceptible (SIS) model on complex networks with imperfect vaccination is proposed. Two types of epidemic spreading patterns (the recovered individuals have or have not immunity) on scale-free networks are discussed. Both theoretical and numerical analyses are presented. The epidemic thresholds related to the vaccination rate, the vaccination-invalid rate and the vaccination success rate on scale-free networks are demonstrated, showing different results from the reported observations. This reveals that whether or not the epidemic can spread over a network under vaccination control is determined not only by the network structure but also by the medicine's effective duration. Moreover, for a given infective rate, the proportion of individuals to vaccinate can be calculated theoretically for the case that the recovered nodes have immunity. Finally, simulated results are presented to show how to control the disease prevalence.