# Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks
## 1 Introduction  
Nowadays, people's lives are closely related to various complex networks, such as social [1], traffic [2] and email [3] networks. Network science is a vast and interdisciplinary research field and is a hot topic in many branches of sciences. Rich-club [4] shows that only a few critical nodes are needed to effectively affect and control the structure and function of the network. Therefore, to identify important nodes is thus significant, allowing us to find influential spreaders [5], control propagation of rumors [6] or epidemics, predict leading figures in academia, conduct advertisements for various commodities [7] and so on. For a variety of specific networks, key node mining can be targeted. In the power network, the protection of important circuit breakers and generating units can effectively prevent the large-scale blackout caused by successive failures [8]. In the application of a search engine, the search results can be sorted according to their matching and importance and returned to the user [9]. When an infectious disease occurs, the source of infection can be treated and isolated in a targeted way to effectively prevent the spread of the infectious agent [10]. In recent years, many critical nodes identification methods in static networks have been proposed [11; 12; 13]. Researchers have aimed to find critical nodes by some heuristic algorithms, such as degree centrality [14], betweenness centrality [15] and k-shell [5]. With the development of deep learning, many researchers are beginning to solve problems in their own fields with the help of deep learning. Representation learning-based [16] methods embed a node as vectors or matrices, then design suitable learning frameworksto learn features of critical nodes, such as RCNN [17], InfGCN [18] and FINDER [19]. All these methods have good performances on various static networks. However, many networks in the real world change over time. Temporal networks are not just an extension of static networks, which contain more in-depth and detailed information [20; 21]. In temporal networks, the identification of critical nodes is not a trivial task. We can easily get the critical nodes in the current network by static methods, but we have no way of knowing whether the current critical nodes are still important in the future. Vital nodes can be identified in static networks, but can only be predicted in temporal networks. At present, there are two main research topics in this area: (1) methods based on structure and propagation dynamics [22]; (2) methods derived from dynamic graph neural networks (DGNNs) [23]. In order to exploit both structured data and temporal information, a new learning framework named the dynamic graph convolutional network (DGCN) is proposed. DGCN converts the problem of finding critical nodes in temporal networks to regression in deep learning. Our approach is based on the special graph convolutional network (GCN) and long short-term memory network (LSTM) [24]. GCN is a type of GNN which has roots in the convolutional neural network (CNN) [25] and is usually used to extract all levels of graph representation and perform graph classification tasks. LSTM is a special type of neural network with a hidden layer that recurs over time and is often used to process sequence data such as stock data. By combining the structural features learned by the special graph convolutional network with the temporal features learned by LSTM, DGCN can predict the node which has might have a strong spreading ability in the future. The performance of DGCN is compared with node2 vec [26], struc2 vec [27], temporal dynamics-sensitive centrality (TDC) [28] and temporal k-shell(TK) [29], by a weighted SIR model [30] on four real-world temporal networks. Experimental results show that DGCN can effectively predict nodes with the best spreading ability, and significantly outperforms benchmark methods in terms of Kendall \(\tau\) coefficient and top \(k\) hit rate.