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Measuring node centrality is important for a wealth of applications, such as influential people identification, information promotion and traffic congestion prevention. Although there are many researches of node centrality proved, most of them have assumed that networks are static. However, many networks in our real life are dynamic, and the edges will appear or disappear over time. Temporal network could describe the interaction order and relationship among network nodes more accurately. It is of more important theoretical and more practical significance to construct proper temporal network model and identify vital nodes. In this paper, by taking into account the coupling strength between different network layers, we present a method, namely similarity-based supra-adjacency matrix (SSAM) method, to represent temporal network and further measure node importance. For a temporal network with N nodes and T layers, the SSAM is a matrix of size NTNT with a collection of both intra-layer relationship and inter-layer relationship. We restrict our attention to inter-layer coupling. Regarding the traditional method of measuring the node similarity of nearest-neighbor layers as one constant value, the neighbor topological overlap information is used to measure the node similarity for the nearest-neighbor layers, which ensures that the couplings of different nodes of inter-layer relationship are different. We then compute the node importance for temporal network based on eigenvector centrality, the dominant eigenvector of similarity-based supra-adjacency matrix, which indicates not only the node i's importance in layer t but also the changing trajectory of the node i's importance across the time. To evaluate the ranking effect of node importance obtained by eigenvector-based centrality, we also study the network robustness and calculate the difference of temporal global efficiency with node deletion approach in this work. In order to compare with the traditional method, we measure the node ranking effect of different time layers by the Kendall rank correlation coefficient of eigenvector centrality and the difference of temporal global efficiency. According to the empirical results on the workspace and Enrons datasets for both SSAM method and tradition method, the SSAM method with neighbor topological overlap information, which takes into account the inter-layer similarity, can effectively avoid overestimating or underestimating the importance of nodes compared with traditional method with one constant value. Furthermore, the experiments for the two datasets show that the average Kendall's could be improved by 17.72% and 12.44% for each layer network, which indicates that the node similarity for different layers is significant to construct temporal network and measure the node importance in temporal network.
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Keywords:
- temporal network /
- inter-layer similarity /
- eigenvector-based centrality /
- temporal global efficiency
[1] Holme P, Saramki J 2013 Temporal Networks (Heidelberg:Springer) pp1-2
[2] Holme P, Saramki J 2012 Phys. Rep. 519 97
[3] Holme P 2015 Eur. Phys. J. B 88 234
[4] Liu J G, Ren Z M, Guo Q 2013 Physica A 392 4154
[5] Ren Z M, Zeng A, Chen D B, Liao H, Liu J G 2014 EPL 106 48005
[6] Liu J G, Ren Z M, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 178901 (in Chinese)[刘建国, 任卓明, 郭强, 汪秉宏 2013 物理学报 62 178901]
[7] L L Y, Chen D B, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1
[8] Ren Z M, Shao F, Liu J G, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 128901 (in Chinese)[任卓明, 邵凤, 刘建国, 郭强, 汪秉宏 2013 物理学报 62 128901]
[9] Liu J G, Lin J H, Guo Q, Zhou T 2016 Sci. Rep. 6 21380
[10] Zhang Y Q, Cui J, Zhang S M, Zhang Q, Li X 2016 Eur. Phys. J. B 89 26
[11] Tang J, Musolesi M, Mascolo C, Latora V 2009 Proceedings of the 2nd ACM Workshop on Online Social Networks Barcelona, Spain, August 17-17, 2009 p31
[12] Tang J, Scellato S, Musolesi M, Mascolo C, Latora V 2010 Phys. Rev. E 81 055101
[13] Deng D M, Zhu J, Chen D B, Gao H 2013 Comput. Sci. 40 26 (in Chinese)[邓冬梅, 朱建, 陈端兵, 高辉 2013 计算机科学 40 26]
[14] Deng D M 2014 M. S. Dissertation (Chengdu:University of Electronic Science and Technology of China) (in Chinese)[邓冬梅 2014 硕士学位论文 (成都:电子科技大学)]
[15] Kim H, Anderson R 2012 Phys. Rev. E 85 026107
[16] Huang D W, Yu Z G 2017 Sci. Rep. 7 41454
[17] Taylor D, Myers S A, Clauset A, Porter M A 2017 Multiscale Model. Simul. 15 537
[18] Zhu Y X, Zhang F L, Qin Z G 2014 J. Comput. Appl. 34 3184 (in Chinese)[朱义鑫, 张凤荔, 秦志光 2014 计算机应用 34 3184]
[19] Gnois M, Vestergaard C L, Fournet J, Panisson A 2015 Network Sci. 3 326
[20] Klimt B, Yang Y 2004 Machine Learning:ECML 2004 217
[21] Zhang Z K, Liu C, Zhan X X, Lu X, Zhang C X, Zhang Y C 2016 Phys. Rep. 651 1-34
[22] Liu C, Zhan X X, Zhang Z K, Sun G Q, Hui P M 2015 New J. Phys. 17 113045
[23] Liu C, Zhang Z K 2014 Commun. Nonlinear Sci. Numerical Simulat. 19 896
[24] Kendall M G 1938 Biometrika 30 81
[25] Agresti A 2010 Analysis of Ordinal Categorical Data (2nd Ed.) (New York:John Wiley Sons John Wiley Sons) pp188-191
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[1] Holme P, Saramki J 2013 Temporal Networks (Heidelberg:Springer) pp1-2
[2] Holme P, Saramki J 2012 Phys. Rep. 519 97
[3] Holme P 2015 Eur. Phys. J. B 88 234
[4] Liu J G, Ren Z M, Guo Q 2013 Physica A 392 4154
[5] Ren Z M, Zeng A, Chen D B, Liao H, Liu J G 2014 EPL 106 48005
[6] Liu J G, Ren Z M, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 178901 (in Chinese)[刘建国, 任卓明, 郭强, 汪秉宏 2013 物理学报 62 178901]
[7] L L Y, Chen D B, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1
[8] Ren Z M, Shao F, Liu J G, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 128901 (in Chinese)[任卓明, 邵凤, 刘建国, 郭强, 汪秉宏 2013 物理学报 62 128901]
[9] Liu J G, Lin J H, Guo Q, Zhou T 2016 Sci. Rep. 6 21380
[10] Zhang Y Q, Cui J, Zhang S M, Zhang Q, Li X 2016 Eur. Phys. J. B 89 26
[11] Tang J, Musolesi M, Mascolo C, Latora V 2009 Proceedings of the 2nd ACM Workshop on Online Social Networks Barcelona, Spain, August 17-17, 2009 p31
[12] Tang J, Scellato S, Musolesi M, Mascolo C, Latora V 2010 Phys. Rev. E 81 055101
[13] Deng D M, Zhu J, Chen D B, Gao H 2013 Comput. Sci. 40 26 (in Chinese)[邓冬梅, 朱建, 陈端兵, 高辉 2013 计算机科学 40 26]
[14] Deng D M 2014 M. S. Dissertation (Chengdu:University of Electronic Science and Technology of China) (in Chinese)[邓冬梅 2014 硕士学位论文 (成都:电子科技大学)]
[15] Kim H, Anderson R 2012 Phys. Rev. E 85 026107
[16] Huang D W, Yu Z G 2017 Sci. Rep. 7 41454
[17] Taylor D, Myers S A, Clauset A, Porter M A 2017 Multiscale Model. Simul. 15 537
[18] Zhu Y X, Zhang F L, Qin Z G 2014 J. Comput. Appl. 34 3184 (in Chinese)[朱义鑫, 张凤荔, 秦志光 2014 计算机应用 34 3184]
[19] Gnois M, Vestergaard C L, Fournet J, Panisson A 2015 Network Sci. 3 326
[20] Klimt B, Yang Y 2004 Machine Learning:ECML 2004 217
[21] Zhang Z K, Liu C, Zhan X X, Lu X, Zhang C X, Zhang Y C 2016 Phys. Rep. 651 1-34
[22] Liu C, Zhan X X, Zhang Z K, Sun G Q, Hui P M 2015 New J. Phys. 17 113045
[23] Liu C, Zhang Z K 2014 Commun. Nonlinear Sci. Numerical Simulat. 19 896
[24] Kendall M G 1938 Biometrika 30 81
[25] Agresti A 2010 Analysis of Ordinal Categorical Data (2nd Ed.) (New York:John Wiley Sons John Wiley Sons) pp188-191
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