The robustness of complex networks against node failure and malicious attack has been of interest for decades, while most of the research has focused on random attack or hub-targeted attack. In many real-world scenarios, however, attacks are neither random nor hub-targeted, but localized, where a group of neighboring nodes in a network are attacked and fail. In this paper we develop a percolation framework to analytically and numerically study the robustness of complex networks against such localized attack. In particular, we investigate this robustness in Erdős-Rényi networks, random-regular networks, and scale-free networks. Our results provide insight into how to better protect networks, enhance cybersecurity, and facilitate the design of more robust infrastructures.The functioning of complex networks such as the internet, airline routes, and social networks is crucially dependent upon the interconnections between network nodes. These interconnections are such that when some nodes in the network fail, others connected through them to the network will also be disabled and the entire network may collapse. In order to understand network robustness and design resilient complex systems, one needs to know whether a complex network can continue to function after a fraction of its nodes have been removed either through node failure or malicious attack . This question is dealt within percolation theory [21][22][23][24] in which the percolation phase transition occurs at some critical occupation probability p c . Above p c , a giant component, defined as a cluster whose size is proportional to that of the entire network, exists; below p c the giant component is absent and the entire network collapses. Only nodes in the giant component continue to function after the node-removal process.The robustness of complex networks under attack is dependent upon the structure of the underlying network and the nature of the attack. Previous research has focused on two types of initial attack: random attack and hub-targeted attack. In a random attack each node in the network is attacked with the same probability [1-3, 8, 10, 21]. In a hub-targeted attack the probability that high-degree nodes will be attacked is higher than that for low-degree nodes [1,3,4,7,12]. An important feature of the network structure is its degree distribution, P(k), which describes the probability that a node has a specific degree k. Networks with different degree distributions behave very differently under different types of attack. For instance, the internet, which shows a power law degree distribution, is extremely robust against random attack but vulnerable to hub-targeted attack [1,4].However these two types of attack-random attack and hub-targeted attack-do not adequately describe many real-world scenarios in which complex networks suffer from damage that is localized, i.e., a node is affected, then its neighbors, and then their neighbors, and so on (see figure 1). Examples include the effects of earthquakes, floods, or military attacks on infrastructu...
-It was recently found that cascading failures can cause the abrupt breakdown of a system of interdependent networks. Using the percolation method developed for single clustered networks by Newman [Phys. Rev. Lett. 103, 058701 (2009)], we develop an analytical method for studying how clustering within the networks of a system of interdependent networks affects the system's robustness. We find that clustering significantly increases the vulnerability of the system, which is represented by the increased value of the percolation threshold pc in interdependent networks.
Clustering, or transitivity has been observed in real networks and its effects on their structure and function has been discussed extensively. The focus of these studies has been on clustering of single networks while the effect of clustering on the robustness of coupled networks received very little attention. Only the case of a pair of fully coupled networks with clustering has been studied recently. Here we generalize the study of clustering of a fully coupled pair of networks to the study of partially interdependent network of networks with clustering within the network components. We show both analytically and numerically, how clustering within the networks, affects the percolation properties of interdependent networks, including percolation threshold, size of giant component and critical coupling point where first order phase transition changes to second order phase transition as the coupling between the networks reduces. We study two types of clustering: one type proposed by Newman [25] where the average degree is kept constant while changing the clustering and the other proposed by Hackett et al. [38] where the degree distribution is kept constant. The first type of clustering is treated both analytically and numerically while the second one is treated only numerically.
The stability of networks is greatly influenced by their degree distributions and in particular by their breadth. Networks with broader degree distributions are usually more robust to random failures but less robust to localized attacks. To better understand the effect of the breadth of the degree distribution we study two models in which the breadth is controlled and compare their robustness against localized attacks (LA) and random attacks (RA). We study analytically and by numerical simulations the cases where the degrees in the networks follow a bi-Poisson distribution, P(k)=αe^{-λ_{1}}λ_{1}^{k}/k!+(1-α)e^{-λ_{2}}λ_{2}^{k}/k!,α∈[0,1], and a Gaussian distribution, P(k)=Aexp(-(k-μ)^{2}/2σ^{2}), with a normalization constant A where k≥0. In the bi-Poisson distribution the breadth is controlled by the values of α, λ_{1}, and λ_{2}, while in the Gaussian distribution it is controlled by the standard deviation, σ. We find that only when α=0 or α=1, i.e., degrees obeying a pure Poisson distribution, are LA and RA the same. In all other cases networks are more vulnerable under LA than under RA. For a Gaussian distribution with an average degree μ fixed, we find that when σ^{2} is smaller than μ the network is more vulnerable against random attack. When σ^{2} is larger than μ, however, the network becomes more vulnerable against localized attack. Similar qualitative results are also shown for interdependent networks.
Photovoltaic (PV) power station faults in the natural environment mainly occur in the PV array, and the accurate fault diagnosis is of particular significance for the safe and efficient PV power plant operation. The PV array's electrical behavior characteristics under fault conditions is analyzed in this paper, and a novel PV array fault diagnosis method is proposed based on fuzzy C-mean (FCM) and fuzzy membership algorithms. Firstly, clustering analysis of PV array fault samples is conducted using the FCM algorithm, indicating that there is a fixed relationship between the distribution characteristics of cluster centers and the different fault, then the fault samples are classified effectively. The membership degrees of all fault data and cluster centers are then determined by the fuzzy membership algorithm for the final fault diagnosis. Simulation analysis indicated that the diagnostic accuracy of the proposed method was 96%. Field experiments further verified the correctness and effectiveness of the proposed method. In this paper, various types of fault distribution features are effectively identified by the FCM algorithm, whether the PV array operation parameters belong to the fault category is determined by fuzzy membership algorithm, and the advantage of the proposed method is it can classify the fault data from normal operating data without foreknowledge.
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