All-Terminal Network Reliability Estimation with Graph Neural Networks

All-terminal network reliability is a crucial feature for critical infrastructures such as transportation, computer, and communication networks. Exact all-terminal network reliability (EATNR) calculation is a non-deterministic polynomial (NP hard) and computationally expensive problem, which has led the development of approximate methods based on artificial neural networks (ANNs). Although there have been some research studies on all-terminal reliability estimation using traditional ANNs, they require the upper bound network reliability as input. More recent deep learning (DL) approaches, such as convolutional neural networks (CNNs), remain limited because networks have varying structures and/or sizes. Deep neural networks (DNNs) have been employed as well. However, DNNs require preprocessing procedures that compress the networks information in vectorial representations, e.g., graph embedding methods (GEMs). As an alternative to address these challenges, this study presents a preliminary development of all-terminal network reliability estimation based on graph neural networks (GNNs), a state-of-the-art deep learning technique. GNNs are appropriate for network reliability prediction as they have the capability of making graph-level regression by directly taking the input information from the graph domain in the form of adjacency matrix, and nodes/links features. To use GNNs for all-terminal network reliability estimation, an applicable architecture of GNN is developed. The network adjacency matrix along with topological attributes such as the node degree, clustering coefficient, and network efficiency are considered as node features that shape the inputs for GNNs. In addition, the proposed architecture guarantees the (reliability) output range [0, 1]. In summary, All-Terminal Network Reliability Estimation with GNN, has advantages over traditional ANNs, CNNs, and DNNs. First, traditional ANNs require the upper bound network reliability as input, an additional time-consuming calculation that GNNs do not need. Second, unlike CNNs, GNNs are able to predict the reliability for networks of varying sizes. Third, whereas DNNs require preprocessing methods to convert the adjacency matrix to vectors, GNNs do not, as they can directly capture the input information from the graph domain. Moreover, the experimental results obtained in the present study show enhanced prediction performance. A six-step process is proposed to exploit GNN capabilities by employing the graph domain information of a random set of networks together with the exact all terminal network reliability (EATNR) as target output during training. Different GNN architectures are investigated. The best input-architecture configuration is selected based on the root mean squared error (RMSE) considering cross-validation. A dataset of 6000 Erdős-Rényi random graphs was generated to this end. The best GNN proposed here, based on the RMSE (0.00934), outperforms previous ANN, CNN, and DNN approaches. The GNN achieved a correlation coefficient of 0.9979 and a p-value of 0.28 for a paired t-test between the actual and predicted reliability. There is also a significant computation time reduction achieved by using GNN (40 ms/network).