Please note this is a comparison between Version 2 by Alfred Zheng and Version 1 by Duanbing Chen.

Many real-world systems can be expressed in temporal networks with nodes playing different roles in structure and function, and edges representing the relationships between nodes. Identifying critical nodes can help us control the spread of public opinions or epidemics, predict leading figures in academia, conduct advertisements for various commodities and so on.Β

- temporal networks
- critical nodes
- node embedding
- deep learning

Nowadays, peopleβs 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]^{[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 frameworks to learn features of critical nodes, such as RCNN ^{[17]}, InfGCN ^{[18]} and FINDER ^{[19]}. All these methods have good performances on various static networks.

- Weng, J.; Lim, E.P.; Jiang, J.; He, Q. TwitterRank: Finding topic-sensitive influential twitterers. In Proceedings of the WSDM 2010-Proceedings of the 3rd ACM International Conference on Web Search and Data Mining, New York, NY, USA, 3β6 February 2010; pp. 261β270.
- Ghosh, S.; Banerjee, A.; Sharma, N.; Agarwal, S.; Ganguly, N.; Bhattacharya, S.; Mukherjee, A. Statistical analysis of the Indian Railway Network: A complex network approach. Acta Phys. Pol. B Proc. Suppl. 2011, 4, 123β137.
- GuimerΓ , R.; Danon, L.; DΓaz-Guilera, A.; Giralt, F.; Arenas, A. Self-similar community structure in a network of human interactions. Phys. Rev. E-Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 2003, 68, 065103.
- Colizza, V.; Flammini, A.; Serrano, M.A.; Vespignani, A. Detecting rich-club ordering in complex networks. Nat. Phys. 2006, 2, 110β115.
- Gallos, L.; Havlin, S.; Kitsak, M.; Liljeros, F.; Makse, H.; Muchnik, L.; Stanley, H. Identification of influential spreaders in complex networks. Nat. Phys. 2010, 6, 888β893.
- Zhou, F.; LΓΌ, L.; Mariani, M.S. Fast influencers in complex networks. Commun. Nonlinear Sci. Numer. Simul. 2019, 74, 69β83.
- Zhang, T.; Li, P.; Yang, L.X.; Yang, X.; Tang, Y.Y.; Wu, Y. A discount strategy in word-of-mouth marketing. Commun. Nonlinear Sci. Numer. Simul. 2019, 74, 167β179.
- Wang, S.; Lv, W.; Zhang, J.; Luan, S.; Chen, C.; Gu, X. Method of power network critical nodes identification and robustness enhancement based on a cooperative framework. Reliab. Eng. Syst. Saf. 2021, 207, 107313.
- Joodaki, M.; Dowlatshahi, M.B.; Joodaki, N.Z. An ensemble feature selection algorithm based on PageRank centrality and fuzzy logic. Knowl.-Based Syst. 2021, 233, 107538.
- Wei, X.; Zhao, J.; Liu, S.; Wang, Y. Identifying influential spreaders in complex networks for disease spread and control. Sci. Rep. 2022, 12, 5550.
- LΓΌ, L.; Chen, D.; Ren, X.L.; Zhang, Q.M.; Zhang, Y.C.; Zhou, T. Vital nodes identification in complex networks. Phys. Rep. 2016, 650, 1β63.
- Guo, C.; Yang, L.; Chen, X.; Chen, D.; Gao, H.; Ma, J. Influential nodes identification in complex networks via information entropy. Entropy 2020, 22, 242.
- Chen, D.B.; Sun, H.L.; Tang, Q.; Tian, S.Z.; Xie, M. Identifying influential spreaders in complex networks by propagation probability dynamics. Chaos 2019, 29, 030120.
- Bonacich, P. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol. 1972, 2, 113β130.
- Freeman, L.C. A set of measures of centrality based on betweenness. Sociometry 1977, 40, 35β41.
- Cui, P.; Wang, X.; Pei, J.; Zhu, W. A Survey on Network Embedding. IEEE Trans. Knowl. Data Eng. 2019, 31, 833β852.
- Yu, E.Y.; Wang, Y.P.; Fu, Y.; Chen, D.B.; Xie, M. Identifying critical nodes in complex networks via graph convolutional networks. Knowl.-Based Syst. 2020, 198, 105893.
- Zhao, G.; Jia, P.; Zhou, A.; Zhang, B. InfGCN: Identifying influential nodes in complex networks with graph convolutional networks. Neurocomputing 2020, 414, 18β26.
- Fan, C.; Zeng, L.; Sun, Y.; Liu, Y.Y. Finding key players in complex networks through deep reinforcement learning. Nat. Mach. Intell. 2020, 2, 317β324.
- Schmidhuber, J. Deep Learning in neural networks: An overview. Neural Netw. 2015, 61, 85β117.
- Schuster, M.; Paliwal, K.K. Bidirectional recurrent neural networks. IEEE Trans. Signal Process. 1997, 45, 2673β2681.
- Hochreiter, S.; Urgen Schmidhuber, J. Long Shortterm Memory. Neural Comput. 1997, 9, 1735β1780.
- Zhang, Z.; Cui, P.; Zhu, W. Deep Learning on Graphs: A Survey. IEEE Trans. Knowl. Data Eng. 2020, 34, 249β270.
- Kim, H.; Anderson, R. Temporal node centrality in complex networks. Phys. Rev. E-Stat. Nonlinear Soft Matter Phys. 2012, 85, 026107.
- Huang, D.W.; Yu, Z.G. Dynamic-Sensitive centrality of nodes in temporal networks. Sci. Rep. 2017, 7, 41454.
- Liu, J.G.; Lin, J.H.; Guo, Q.; Zhou, T. Locating influential nodes via dynamics-sensitive centrality. Sci. Rep. 2016, 6, 21380.
- Taylor, D.; Myers, S.A.; Clauset, A.; Porter, M.A.; Mucha, P.J. Eigenvector-based centrality measures for temporal networks. Multiscale Model. Simul. 2017, 15, 537β574.
- Huang, Q.; Zhao, C.; Zhang, X.; Wang, X.; Yi, D. Centrality measures in temporal networks with time series analysis. Epl 2017, 118, 36001.
- Chandran, J.; Viswanatham, V.M. Dynamic node influence tracking based influence maximization on dynamic social networks. Microprocess. Microsyst. 2022, 95, 104689.
- Jiang, J.L.; Fang, H.; Li, S.Q.; Li, W.M. Identifying important nodes for temporal networks based on the ASAM model. Phys. A Stat. Mech. Its Appl. 2022, 586, 126455.
- Bi, J.; Jin, J.; Qu, C.; Zhan, X.; Wang, G.; Yan, G. Temporal gravity model for important node identification in temporal networks. Chaos Solitons Fractals 2021, 147, 110934.
- Niepert, M.; Ahmad, M.; Kutzkov, K. Learning convolutional neural networks for graphs. In Proceedings of the 33rd International Conference on Machine Learning, ICML 2016, New York, NY, USA, 19β24 June 2016; Volume 4, pp. 2958β2967.
- Kipf, T.N.; Welling, M. Semi-supervised classification with graph convolutional networks. In Proceedings of the 5th International Conference on Learning Representations, ICLR 2017-Conference Track Proceedings, Toulon, France, 24β26 April 2017.
- Grover, A.; Leskovec, J. Node2vec: Scalable feature learning for networks. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, 13β17 August 2016; pp. 855β864.
- Ribeiro, L.F.; Saverese, P.H.; Figueiredo, D.R. Struc2vec: Learning Node Representations from Structural Identity. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Halifax, NS, Canada, 13β17 August 2017; pp. 13β17.
- Kazemi, S.M.; Kobyzev, I.; Forsyth, P.; Goel, R.; Jain, K.; Kobyzev, I.; Sethi, A.; Forsyth, P.; Poupart, P.; Goel, R.; et al. Representation Learning for Dynamic Graphs: A Survey. J. Mach. Learn. Res. 2020, 21, 2648β2720.
- Wu, Z.; Pan, S.; Chen, F.; Long, G.; Zhang, C.; Yu, P.S. A Comprehensive Survey on Graph Neural Networks. IEEE Trans. Neural Netw. Learn. Syst. 2021, 32, 4β24.
- Medsker, L.R.; Jain, L.C. Recurrent Neural Networks Design and Applications. J. Chem. Inf. Model. 2013, 53, 1689β1699.
- Seo, Y.; Defferrard, M.; Vandergheynst, P.; Bresson, X. Structured sequence modeling with graph convolutional recurrent networks. In Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); LNCS; Springer: Berlin/Heidelberg, Germany, 2018; Volume 11301, pp. 362β373.
- Defferrard, M.; Bresson, X.; Vandergheynst, P. Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering. Adv. Neural Inf. Process. Syst. 2016, 59, 395β398.
- Taheri, A.; Gimpel, K.; Berger-Wolf, T. Learning to represent the evolution of dynamic graphs with recurrent models. In Proceedings of the Web Conference 2019-Companion of the World Wide Web Conference, WWW 2019, San Francisco, CA, USA, 13β17 May 2019; pp. 301β307.
- Li, Y.; Tarlow, D.; Brockschmidt, M.; Zemel, R. Gated Graph Sequence Neural Networks. arXiv 2015, arXiv:1511.05493.
- Chen, J.; Xu, X.; Wu, Y.; Zheng, H. GC-LSTM: Graph convolution embedded LSTM for dynamic link prediction. arXiv 2018, arXiv:1812.04206.
- Munikoti, S.; Das, L.; Natarajan, B. Scalable graph neural network-based framework for identifying critical nodes and links in complex networks. Neurocomputing 2022, 468, 211β221.

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