Network robustness is critical for various industrial and social networks against malicious attacks, which has various meanings in different research contexts and here it refers to the ability of a network to sustain its functionality when a fraction of the network fail to work due to attacks. The rapid development of complex networks research indicates special interest and great concern about the network robustness, which is essential for further analyzing and optimizing network structures towards engineering applications. This comprehensive survey distills the important findings and developments of network robustness research, focusing on the a posteriori structural robustness measures for single-layer static networks. Specifically, the a posteriori robustness measures are reviewed from four perspectives: 1) network functionality, including connectivity, controllability and communication ability, as well as their extensions; 2) malicious attacks, including conventional and computation-based attack strategies; 3) robustness estimation methods using either analytical approximation or machine learning-based prediction; 4) network robustness optimization. Based on the existing measures, a practical threshold of network destruction is introduced, with the suggestion that network robustness should be measured only before reaching the threshold of destruction. Then, a posteriori and a priori measures are compared experimentally, revealing the advantages of the a posteriori measures. Finally, prospective research directions with respect to a posteriori robustness measures are recommended.

Network controllability measures how well a networked system can be controlled to a target state, and its robustness reflects how well the system can maintain the controllability against malicious attacks by means of node removals or edge removals. The measure of network controllability is quantified by the number of external control inputs needed to recover or to retain the controllability after the occurrence of an unexpected attack. The measure of the network controllability robustness, on the other hand, is quantified by a sequence of values that record the remaining controllability of the network after a sequence of attacks. Traditionally, the controllability robustness is determined by attack simulations, which is computationally time consuming. In this article, a method to predict the controllability robustness based on machine learning using a convolutional neural network (CNN) is proposed, motivated by the observations that: 1) there is no clear correlation between the topological features and the controllability robustness of a general network; 2) the adjacency matrix of a network can be regarded as a grayscale image; and 3) the CNN technique has proved successful in image processing without human intervention. Under the new framework, a fairly large number of training data generated by simulations are used to train a CNN for predicting the controllability robustness according to the input network-adjacency matrices, without performing conventional attack simulations. Extensive experimental studies were carried out, which demonstrate that the proposed framework for predicting controllability robustness of different network configurations is accurate and reliable with very low overheads.

Network robustness is critical for various societal and industrial networks against malicious attacks. In particular, connectivity robustness and controllability robustness reflect how well a networked system can maintain its connectedness and controllability against destructive attacks, which can be quantified by a sequence of values that record the remaining connectivity and controllability of the network after a sequence of node- or edge-removal attacks. Traditionally, robustness is determined by attack simulations, which are computationally very time-consuming or even practically infeasible for large-scale networks. In this article, an improved method for network robustness prediction is developed based on learning feature representation using the convolutional neural network (LFR-CNN). In this scheme, the higher-dimensional network data are compressed into lower-dimensional representations, which are then passed to a convolutional neural network to perform robustness prediction. Extensive experimental studies on both synthetic and real-world networks, both directed and undirected, demonstrate that: 1) the proposed LFR-CNN performs better than other two state-of-the-art prediction methods, with significantly smaller prediction errors; 2) LFR-CNN is insensitive to the variation of the input network size, which significantly extends its applicability; 3) although LFR-CNN needs more time to perform feature learning, it can achieve accurate prediction faster than attack simulations; and 4) LFR-CNN not only accurately predicts the network robustness, but also provides a good indicator for connectivity robustness, better than the classical spectral measures.

To quantitatively measure the connectedness robustness of a complex network, a sequence of values that record the remaining connectedness of the network after a sequence of node- or edge-removal attacks can be used. However, it is computationally time-consuming to measure the network connectedness robustness by attack simulations for large-scale networked systems. In the present paper, an efficient method based on convolutional neural network (CNN) is proposed to train for estimating the network connectedness robustness. The new approach is motivated by the facts that 1) the adjacency matrix of a network can be converted to a gray-scale image and CNN is very powerful for image processing, and 2) CNN has proved very effective in predicting the controllability robustness of complex networks. Extensive experimental studies on directed and undirected, as well as synthetic and real-world networks suggest that: 1) the proposed CNN-based methodology performs excellently in the prediction of the connectedness robustness of complex networks as a process; 2) it performs fairly well as the indicator for the connectedness robustness, compared to other predictive measures.

This paper introduces a quantitative measure of the robustness of network controllability, and presents an empirical comparison on 6 network models, i.e., random-graph network, scale-free network, multiplex concurrence network, q-snapback network, random triangle network, and random rectangle network, against 6 different attacks, i.e., betweenness-based and degree-based, random and targeted, node-removal and edge-removal attacks, showing the overall good performances of the multi-ring structure in networks.

Network controllability robustness (CR) reflects how well a networked system can maintain its controllability against destructive attacks. Its measure is quantified by a sequence of values that record the remaining controllability of the network after a sequence of node-removal or edge-removal attacks. Traditionally, the CR is determined by attack simulations, which is computationally time-consuming or even infeasible. In this article, an improved method for predicting the network CR is developed based on machine learning using a group of convolutional neural networks (CNNs). In this scheme, a number of training data generated by simulations are used to train the group of CNNs for classification and prediction, respectively. Extensive experimental studies are carried out, which demonstrate that 1) the proposed method predicts more precisely than the classical single-CNN predictor; 2) the proposed CNN-based predictor provides a better predictive measure than the traditional spectral measures and network heterogeneity.