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.

Despite various measures across different engineering and social systems, network robustness remains crucial for resisting random faults and malicious attacks. In this study, robustness refers to the ability of a network to maintain its functionality after a part of the network has failed. Existing methods assess network robustness using attack simulations, spectral measures, or deep neural networks (DNNs), which return a single metric as a result. Evaluating network robustness is technically challenging, while evaluating a single metric is practically insufficient. This article proposes a multitask analysis system based on the graph isomorphism network (GIN) model, abbreviated as GIN-MAS. First, a destruction-based robustness metric is formulated using the destruction threshold of the examined network. A multitask learning approach is taken to learn the network robustness metrics, including connectivity robustness, controllability robustness, destruction threshold, and the maximum number of connected components. Then, a five-layer GIN is constructed for evaluating the aforementioned four robustness metrics simultaneously. Finally, extensive experimental studies reveal that 1) GIN-MAS outperforms nine other methods, including three state-of-the-art convolutional neural network (CNN)-based robustness evaluators, with lower prediction errors for both known and unknown datasets from various directed and undirected, synthetic, and real-world networks; 2) the multitask learning scheme is not only capable of handling multiple tasks simultaneously but more importantly it enables the parameter and knowledge sharing across tasks, thus preventing overfitting and enhancing the performances; and 3) GIN-MAS performs multitasks significantly faster than other single-task evaluators. The excellent performance of GIN-MAS suggests that more powerful DNNs have great potentials for analyzing more complicated and comprehensive robustness evaluation tasks.

This paper addresses the robustness of a network to sustain its connectivity and controllability against malicious attacks. This kind of network robustness is typically measured by the time-consuming attack simulation, which returns a sequence of values that record the remaining connectivity and controllability after a sequence of node-or edge-removal attacks. For improvement, this paper develops an efficient framework for network robustness prediction, the spatial pyramid pooling convolutional neural network (SPP-CNN). The new framework installs a spatial pyramid pooling layer between the convolutional and fully-connected layers, overcoming the common mismatch issue in the CNN-based prediction approaches and extending its generalizability. Extensive experiments are carried out by comparing SPP-CNN with three state-of-the-art robustness predictors, namely one CNN-based and two graph neural networks-based frameworks. Synthetic and real-world networks, both directed and undirected, are investigated. Experimental results demonstrate that the proposed SPP-CNN achieves better prediction performances and better generalizability for both cases of known and unknown datasets, with significantly lower time-consumption, than its counterparts.

Network robustness refers to the ability of a network to continue its functioning against malicious attacks, which is critical for various natural and industrial networks. Network robustness can be quantitatively measured by a sequence of values that record the remaining functionality after a sequential node-or edge-removal attacks. Robustness evaluations are traditionally determined by attack simulations, which are computationally very time-consuming and sometimes practically infeasible. The convolutional neural network (CNN)-based prediction provides a cost-efficient approach to fast evaluating the network robustness. In this article, the prediction performances of the learning feature representation-based CNN (LFR-CNN) and PATCHY-SAN methods are compared through extensively empirical experiments. Specifically, three distributions of network size in the training data are investigated, including the uniform, Gaussian, and extra distributions. The relationship between the CNN input size and the dimension of the evaluated network is studied. Extensive experimental results reveal that compared to the training data of uniform distribution, the Gaussian and extra distributions can significantly improve both the prediction performance and the generalizability, for both LFR-CNN and PATCHY-SAN, and for various functionality robustness. The extension ability of LFR-CNN is significantly better than PATCHY-SAN, verified by extensive comparisons on predicting the robustness of unseen networks. In general, LFR-CNN outperforms PATCHY-SAN, and thus LFR-CNN is recommended over PATCHY-SAN. However, since both LFR-CNN and PATCHY-SAN have advantages for different scenarios, the optimal settings of the input size of CNN are recommended under different configurations.

Many real-world natural and social systems can be modeled as complex networks. As random failures and malicious attacks can seriously destroy the structure of complex networks, it is critical to ensure their robustness and maintain the functions. Generally, connectivity and controllability robustness are adopted to evaluate the performance of networked systems against external attacks and/or failures. A sequence of values is measured to dynamically indicate the network robustness with iterative node- or edge-removal. Calculating the robustness of large-scale real-world networks is usually time consuming, whereas deep-learning provides an efficient methodology to estimate network robustness performance. In this paper, a multi-convolutional neural network (CNN) method called Real-RP is designed to predict the robustness of real-world complex networks. Unknown real-world networks are first classified into known network categories, and their robustness performance is then predicted based on the knowledge of the specific network category trained using a substantial number of synthetic networks. Experimental results show that: 1) real-world complex networks can be classified by a CNN with high precision, and 2) the robustness performance of real-world networks can be predicted with lower average errors compared to existing methods.