# Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks

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## Abstract

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## 1. Introduction

- Convert the problem of finding critical nodes in temporal networks to regression in deep learning.
- A novel and effective learning framework based on the combination of special GCNs and RNNs is proposed to identify nodes with the best spreading ability.
- The proposed method is superior to both classical and state-of-the-art methods when applied to four real-world temporal networks.
- It takes linear time complexity to deal with large networks with thousands of nodes and edges in a short time.

## 2. Related Works

## 3. Background

#### 3.1. Temporal Networks

#### 3.2. Recurrent Neural Network

#### 3.3. Graph Neural Network

## 4. Method

#### 4.1. Problem Definition

#### 4.2. Dynamic Graph Convolutional Network

Algorithm 1: The flow of DGCN. |

#### 4.3. Complexity Analysis

## 5. Experiments

#### 5.1. Datasets

- Email [53]. The directed temporal network is generated by the mail data from a research institution in Europe.
- Contact [54]. An undirected temporal network contains connections between users using mobile wireless devices. An edge is generated if two people are in contact.
- DNC [55]. A directed temporal network of emails in the 2016 Democratic National Committee email leak.
- UCI [56]. A directed temporal network contains sent messages between students at the University of California, Irvine.

#### 5.2. Experimental Settings

#### 5.3. Benchmark Methods

#### 5.4. Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- 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, 3–6 February 2010; pp. 261–270. [Google Scholar]
- 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, 123–137. [Google Scholar] [CrossRef] - Guimerà, R.; Danon, L.; Díaz-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. [Google Scholar] [CrossRef] [Green Version] - Colizza, V.; Flammini, A.; Serrano, M.A.; Vespignani, A. Detecting rich-club ordering in complex networks. Nat. Phys.
**2006**, 2, 110–115. [Google Scholar] [CrossRef] [Green Version] - 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, 888–893. [Google Scholar] - Zhou, F.; Lü, L.; Mariani, M.S. Fast influencers in complex networks. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 74, 69–83. [Google Scholar] [CrossRef] [Green Version] - 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, 167–179. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Wei, X.; Zhao, J.; Liu, S.; Wang, Y. Identifying influential spreaders in complex networks for disease spread and control. Sci. Rep.
**2022**, 12, 5550. [Google Scholar] [CrossRef] - Lü, L.; Chen, D.; Ren, X.L.; Zhang, Q.M.; Zhang, Y.C.; Zhou, T. Vital nodes identification in complex networks. Phys. Rep.
**2016**, 650, 1–63. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] - Bonacich, P. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol.
**1972**, 2, 113–130. [Google Scholar] [CrossRef] - Freeman, L.C. A set of measures of centrality based on betweenness. Sociometry
**1977**, 40, 35–41. [Google Scholar] [CrossRef] - Cui, P.; Wang, X.; Pei, J.; Zhu, W. A Survey on Network Embedding. IEEE Trans. Knowl. Data Eng.
**2019**, 31, 833–852. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] - Zhao, G.; Jia, P.; Zhou, A.; Zhang, B. InfGCN: Identifying influential nodes in complex networks with graph convolutional networks. Neurocomputing
**2020**, 414, 18–26. [Google Scholar] [CrossRef] - Fan, C.; Zeng, L.; Sun, Y.; Liu, Y.Y. Finding key players in complex networks through deep reinforcement learning. Nat. Mach. Intell.
**2020**, 2, 317–324. [Google Scholar] [CrossRef] - Holme, P.; Saramäki, J. Temporal networks. Phys. Rep.
**2012**, 519, 97–125. [Google Scholar] [CrossRef] [Green Version] - Michail, O.; Spirakis, P. Elements of the theory of dynamic networks. Commun. ACM
**2018**, 61, 72. [Google Scholar] [CrossRef] - Pan, R.K.; Saramäki, J. Path lengths, correlations, and centrality in temporal networks. Phys. Rev. E-Stat. Nonlinear Soft Matter Phys.
**2011**, 84, 1577–1589. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Skarding, J.; Gabrys, B.; Musial, K. Foundations and modelling of dynamic networks using Dynamic Graph Neural Networks: A survey. arXiv
**2020**, arXiv:2005.07496. [Google Scholar] - Gers, F.A.; Schraudolph, N.N.; Schmidhuber, J. Learning precise timing with LSTM recurrent networks. J. Mach. Learn. Res.
**2003**, 3, 115–143. [Google Scholar] - LeCun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. Gradient-based learning applied to document recognition. Proc. IEEE
**1998**, 86, 2278–2323. [Google Scholar] [CrossRef] [Green Version] - 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, 13–17 August 2016; pp. 855–864. [Google Scholar]
- 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, 13–17 August 2017; pp. 13–17. [Google Scholar]
- Huang, D.W.; Yu, Z.G. Dynamic-Sensitive centrality of nodes in temporal networks. Sci. Rep.
**2017**, 7, 41454. [Google Scholar] [CrossRef] [Green Version] - Ye, Z.; Zhan, X.; Zhou, Y.; Liu, C.; Zhang, Z.K. Identifying vital nodes on temporal networks: An edge-based K-shell decomposition. In Proceedings of the Chinese Control Conference, CCC, Dalian, China, 26–28 July 2017; pp. 1402–1407. [Google Scholar]
- Kimura, M.; Saito, K.; Motoda, H. Blocking links to minimize contamination spread in a social network. ACM Trans. Knowl. Discov. Data
**2009**, 3, 1–23. [Google Scholar] [CrossRef] - Kim, H.; Anderson, R. Temporal node centrality in complex networks. Phys. Rev. E-Stat. Nonlinear Soft Matter Phys.
**2012**, 85, 026107. [Google Scholar] [CrossRef] [Green Version] - Liu, J.G.; Lin, J.H.; Guo, Q.; Zhou, T. Locating influential nodes via dynamics-sensitive centrality. Sci. Rep.
**2016**, 6, 21380. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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, 537–574. [Google Scholar] [CrossRef] [Green Version] - Huang, Q.; Zhao, C.; Zhang, X.; Wang, X.; Yi, D. Centrality measures in temporal networks with time series analysis. Epl
**2017**, 118, 36001. [Google Scholar] [CrossRef] - Chandran, J.; Viswanatham, V.M. Dynamic node influence tracking based influence maximization on dynamic social networks. Microprocess. Microsyst.
**2022**, 95, 104689. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - 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, 19–24 June 2016; Volume 4, pp. 2958–2967. [Google Scholar]
- 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, 24–26 April 2017. [Google Scholar]
- 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, 2648–2720. [Google Scholar] - 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, 4–24. [Google Scholar] [CrossRef] [Green Version] - Medsker, L.R.; Jain, L.C. Recurrent Neural Networks Design and Applications. J. Chem. Inf. Model.
**2013**, 53, 1689–1699. [Google Scholar] - 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. 362–373. [Google Scholar]
- Defferrard, M.; Bresson, X.; Vandergheynst, P. Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering. Adv. Neural Inf. Process. Syst.
**2016**, 59, 395–398. [Google Scholar] - 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, 13–17 May 2019; pp. 301–307. [Google Scholar]
- Li, Y.; Tarlow, D.; Brockschmidt, M.; Zemel, R. Gated Graph Sequence Neural Networks. arXiv
**2015**, arXiv:1511.05493. [Google Scholar] - Chen, J.; Xu, X.; Wu, Y.; Zheng, H. GC-LSTM: Graph convolution embedded LSTM for dynamic link prediction. arXiv
**2018**, arXiv:1812.04206. [Google Scholar] [CrossRef] - Munikoti, S.; Das, L.; Natarajan, B. Scalable graph neural network-based framework for identifying critical nodes and links in complex networks. Neurocomputing
**2022**, 468, 211–221. [Google Scholar] [CrossRef] - Schmidhuber, J. Deep Learning in neural networks: An overview. Neural Netw.
**2015**, 61, 85–117. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schuster, M.; Paliwal, K.K. Bidirectional recurrent neural networks. IEEE Trans. Signal Process.
**1997**, 45, 2673–2681. [Google Scholar] [CrossRef] [Green Version] - Hochreiter, S.; Urgen Schmidhuber, J. Long Shortterm Memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] - Zhang, Z.; Cui, P.; Zhu, W. Deep Learning on Graphs: A Survey. IEEE Trans. Knowl. Data Eng.
**2020**, 34, 249–270. [Google Scholar] [CrossRef] [Green Version] - Paranjape, A.; Benson, A.R.; Leskovec, J. Motifs in temporal networks. In Proceedings of the WSDM 2017-Proceedings of the 10th ACM International Conference on Web Search and Data Mining, Cambridge, UK, 6–10 February 2017; pp. 601–610. [Google Scholar]
- Chaintreau, A.; Hui, P.; Crowcroft, J.; Diot, C.; Gass, R.; Scott, J. Impact of human mobility on opportunistic forwarding algorithms. IEEE Trans. Mob. Comput.
**2007**, 6, 606–620. [Google Scholar] [CrossRef] - Yu, E.Y.; Fu, Y.; Chen, X.; Xie, M.; Chen, D.B. Identifying critical nodes in temporal networks by network embedding. Sci. Rep.
**2020**, 10, 12494. [Google Scholar] [CrossRef] - Opsahl, T.; Panzarasa, P. Clustering in weighted networks. Soc. Netw.
**2009**, 31, 155–163. [Google Scholar] [CrossRef] - Knight, W.R. A Computer Method for Calculating Kendall’s Tau with Ungrouped Data. J. Am. Stat. Assoc.
**1966**, 61, 436. [Google Scholar] [CrossRef]

**Figure 2.**Unweighted snapshots and weighted snapshots: (

**a**) Unweighted snapshots; (

**b**) weighted snapshots, $\delta =1$.

**Table 1.**Statistical properties of four real-world networks. N and M are the total number of nodes and edges, respectively. $\delta $ is the time interval (hours) used to divide networks and L is the number of snapshots.

Networks | N | M | $\mathit{\delta}$ | L |
---|---|---|---|---|

986 | 332,334 | 168 | 75 | |

Contact | 274 | 28,244 | 1 | 69 |

DNC | 1865 | 39,623 | 12 | 56 |

UCI | 1899 | 59,835 | 24 | 58 |

Networks | $\mathit{s}=1$ | $\mathit{s}=2$ | $\mathit{s}=4$ | $\mathit{s}=8$ | $\mathit{s}=15$ | $\mathit{s}=30$ |
---|---|---|---|---|---|---|

69.17 | 77.45 | 94.46 | 132.30 | 195.10 | 341.90 | |

Contact | 21.54 | 22.73 | 28.06 | 38.87 | 58.55 | 106.61 |

DNC | 128.01 | 147.25 | 175.50 | 247.95 | 362.78 | 643.84 |

UCI | 126.32 | 142.78 | 176.30 | 261.63 | 385.58 | 752.10 |

Networks | DGCN | N2V-LSTM | S2V-LSTM | TDC | TK |
---|---|---|---|---|---|

2.46 | 0.08 | 0.07 | 26.67 | 0.32 | |

Contact | 0.09 | 0.04 | 0.04 | 1.20 | 0.06 |

DNC | 0.45 | 0.16 | 0.14 | 132.71 | 0.37 |

UCI | 0.23 | 0.17 | 0.13 | 140.74 | 0.36 |

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**MDPI and ACS Style**

Yu, E.; Fu, Y.; Zhou, J.; Sun, H.; Chen, D.
Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks. *Appl. Sci.* **2023**, *13*, 7272.
https://doi.org/10.3390/app13127272

**AMA Style**

Yu E, Fu Y, Zhou J, Sun H, Chen D.
Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks. *Applied Sciences*. 2023; 13(12):7272.
https://doi.org/10.3390/app13127272

**Chicago/Turabian Style**

Yu, Enyu, Yan Fu, Junlin Zhou, Hongliang Sun, and Duanbing Chen.
2023. "Predicting Critical Nodes in Temporal Networks by Dynamic Graph Convolutional Networks" *Applied Sciences* 13, no. 12: 7272.
https://doi.org/10.3390/app13127272