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基于信息熵与迭代因子的复杂网络节点重要性评价方法

汪亭亭 梁宗文 张若曦

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基于信息熵与迭代因子的复杂网络节点重要性评价方法

汪亭亭, 梁宗文, 张若曦

Importance evaluation method of complex network nodes based on information entropy and iteration factor

Wang Ting-Ting, Liang Zong-Wen, Zhang Ruo-Xi
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  • 在复杂网络的研究中, 如何有效地衡量节点的重要性一直都是学者们关心的问题. 在节点重要性研究领域, 基于拓扑学信息来判断节点重要性的方法被大量提出, 如K-shell方法. K-shell是一种寻找可能具有重要影响力节点的有效方法, 在大量的研究工作中被广泛引用. 但是, K-shell过多地强调了中心节点的影响力, 却忽视了处于网络外围节点作用力的影响. 为了更好地衡量网络中各个节点对传播的促进作用, 本文提出了一种基于迭代因子和节点信息熵的改进方法来评估各个层次节点的传播能力. 为评价本文方法的性能, 本文采用SIR模型进行仿真实验来对各节点的传播效率进行评估, 并在实验中将本文算法和其他算法进行了对比. 实验结果表明, 本文所提方法具有更好的性能, 并且适合解决大规模复杂网络中的节点重要性评价问题.
    In the study of complex networks, researchers have long focused on the identification of influencing nodes. Based on topological information, several quantitative methods of determining the importance of nodes are proposed. K-shell is an efficient way to find potentially affected nodes. However, the K-shell overemphasizes the influence of the location of the central nodebut ignores the effect of the force of the nodes located at the periphery of the network. Furthermore, the topology of real networks is complex, which makes the computation of the K-shell problem for large scale-free networks extremely difficult. In order to avoid ignoring the contribution of any node in the network to the propagation, this work proposes an improved method based on the iteration factor and information entropy to estimate the propagation capability of each layer of nodes. This method not only achieves the accuracy of node ordering, but also effectively avoids the phenomenon of rich clubs. To evaluate the performance of this method, the SIR model is used to simulate the propagation efficiency of each node, and the algorithm is compared with other algorithms. Experimental results show that this method has better performance than other methods and is suitable for large-scale networks.
      Corresponding author: Liang Zong-Wen, zongwen-liang@hotmail.com
    [1]

    Pastor-Satorras R, Vespignani A 2002 Phys. Rev. E 65 036104Google Scholar

    [2]

    Leskovec J, Adamic L A, Huberman B A 2007 Acm Trans. Web 1 5Google Scholar

    [3]

    Freeman L C 1978 Soc. Networks 1 215Google Scholar

    [4]

    Freeman L C 1977 Sociometry 40 35Google Scholar

    [5]

    Sabidussi G 1966 Psychometrika 31 581Google Scholar

    [6]

    Lü L Y, Zhou T, Zhang Q M, Stanley H E 2016 Nat. Commun. 7 10168Google Scholar

    [7]

    Lü L Y, Chen D B, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1Google Scholar

    [8]

    Brin S, Page L 1998 Comput. Netw. ISDN Syst. 30 107Google Scholar

    [9]

    Lü L Y, Zhang Y C, Yeung C H, Zhou T 2011 PloS One 6 21202Google Scholar

    [10]

    Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nat. Phys. 6 888Google Scholar

    [11]

    Pei S, Muchnik L, Andrade J S, Zheng Z M, Makse H A 2014 Sci. Rep. 4 5547Google Scholar

    [12]

    Montresor A, De Pellegrini F, Miorandi D 2011 Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing San Jose, CA, June 6–8, 2011 p207

    [13]

    Basaras P, Katsaros D, Tassiulas L 2013 Computer 46 24Google Scholar

    [14]

    Wang Z X, Zhao Y, Xi J K, Du C J 2016 Physica A 461 171Google Scholar

    [15]

    Zhou S, Mondragon R J 2004 IEEE Commun. Lett. 8 180Google Scholar

    [16]

    Wang M, Li W C, Guo Y N, Peng X Y, Li Y X 2020 Physica A 554 124229Google Scholar

    [17]

    Zareie A, Sheikhahmadi A, Jalili M, Fasaei M S K 2020 Knowledge-Based Syst. 194 105580Google Scholar

    [18]

    Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200Google Scholar

    [19]

    Hethcote H W 2000 SIAM Rev. 42 599Google Scholar

    [20]

    Ma L I, Ma C, Zhang H F, Wang B H 2016 Physica A 451 205Google Scholar

    [21]

    Li Z, Ren T, Ma X Q, Liu S M, Zhang Y X, Zhou T 2019 Sci. Rep. 9 8387Google Scholar

    [22]

    Bae J, Kim S 2014 Physica A 395 549Google Scholar

    [23]

    Bhat N, Aggarwal N, Kumar S 2020 Procedia Comput Sci. 171 662Google Scholar

    [24]

    阮逸润, 老松杨, 汤俊, 白亮 2020 物理学报 71 176401Google Scholar

    Ruan Y R, Lao S Y, Tang J, Bai L, Guo Y M 2020 Acta Phys. Sin. 71 176401Google Scholar

    [25]

    Colizza V, Flammini A, Serrano M A, Vespignani A 2006 Nat. Phys. 2 110Google Scholar

    [26]

    Rui X B, Meng F R, Wang Z X, Yuan G 2019 Appl. Intell. 49 2684Google Scholar

    [27]

    Liu D, Jing Y, Zhao J, Wang W J, Song G J 2017 Sci. Rep. 7 43330Google Scholar

    [28]

    Namtirtha A, Dutta A, Dutta B 2018 Physica A 499 310Google Scholar

    [29]

    Kim H, Anderson R 2012 Phys. Rev. E 85 026107Google Scholar

    [30]

    Takaguchi T, Sato N, Yano K, Masuda N 2012 New J. Phys. 14 093003Google Scholar

    [31]

    Qu C Q, Zhan X X, Wang G H, Wu J L, Zhang Z K 2019 Chaos 29 033116Google Scholar

    [32]

    胡钢, 许丽鹏, 徐翔 2021 物理学报 70 108901Google Scholar

    Hu G, Xu L P, Xu X 2021 Acta Phys. Sin. 70 108901Google Scholar

    [33]

    Newman M E J 2006 Phys. Rev. E 74 036104Google Scholar

    [34]

    Yin H, Benson A R, Leskovec J, Gleich D F 2017 Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (Halifax, Candana) August 13–17, 2017 p555

    [35]

    Adamic L A 2005 Glance N Proceedings of the 3rd International Workshop on Link Discovery (New York, USA) 2005 p36

    [36]

    Mcauley J, Leskovec J 2012 Proceedings of the 25th International Conference on Neural Information Processing Systems (Lake Tahoe, Nevada) 2012 p539

    [37]

    Leskovec J, Huttenlocher D, Kleinberg J 2010 Proceedings of the 19th International Conference on World Wide Web (New York, USA) 2010 p65

    [38]

    Rozemberczki B, Davies R, Sarkar R, Sutton C 2019 Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining New York, USA, 2019 p65

    [39]

    Rocha L, Liljeros F, Holme P 2011 PLoS Comput. Biol. 7 1001109Google Scholar

    [40]

    Leskovec J, Kleinberg J, Faloutsos C 2007 ACM Trans. Knowl. Discovery Data 1 2Google Scholar

    [41]

    Moreno Y, Pastor-Satorras R, Vespignani A 2002 Eur. Phys. J. B 26 521Google Scholar

    [42]

    Kenall M G 1938 Biometrika 30 81Google Scholar

    [43]

    Zhang J X, Chen D B, Dong Q, Zhao Z D 2016 Sci. Rep. 6 27823Google Scholar

    [44]

    Morone F, Makse H 2015 Nature 524 65Google Scholar

    [45]

    Goyal A, Lu W, Lakshmanan L 2011 Proceedings of the 20th International Conference on World Wide Web Hyderabad, India, 2011 p47

    [46]

    Jung K, Heo W, Chen W 2012 IEEE 12th International Conference on Data Mining Brussels, Belgium, December 10–13, 2012 p918

  • 图 1  一个简单示例图

    Fig. 1.  A sample graph.

    图 2  不同方法下不同比例源扩散器的平均最短路径长度Ls (a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat

    Fig. 2.  Average shortest path length Ls under different proportion of source spreaders by different methods: (a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat.

    图 3  比较在相同时间内感染节点总数的百分比 (a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat

    Fig. 3.  Compare the percentage of the total number of infected nodes over the same time period: (a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat.

    图 4  比较不同传播时间 t 中前 10% 种子节点感染节点的百分比 (a) NS; (b) Facebook; (c) WV; (d) Sex; (f) Sport; (f) CondMat

    Fig. 4.  Compare the percentage of nodes infected by the top 10% of seed nodes in different propagation time t: (a) NS; (b) Facebook; (c) WV; (d) Sex; (f) Sport; (f) CondMat.

    表 1  节点在每个shell中的信息熵

    Table 1.  Information entropy of each node.

    ksNodeE
    310.9571
    40.8565
    50.8099
    20.7151
    30.6366
    27/8/90.4861
    60.4374
    1210.6675
    100.4034
    230.3720
    20/22/24/25/260.2420
    11/12/13/14/160.1992
    150.1733
    17/18/190.1435
    下载: 导出CSV

    表 2  节点在每个迭代层中的信息熵

    Table 2.  Information entropy of each node.

    IterationNodeE+
    751.3728
    611.4579
    41.4378
    21.2159
    31.0589
    57/80.7839
    460.7189
    90.6430
    3210.8084
    2100.4818
    230.3720
    1160.3400
    150.3139
    20/22/24/25/260.2420
    170.2219
    11/12/13/140.1992
    18/190.1435
    下载: 导出CSV
    IE+算法伪代码
    输入: 网络结构G =(V, E)
    输出: 网络中节点的排序索引Rank
    1: 通过G = (V, E)得出邻接矩阵A
    2: 通过K-shell算法得出每个节点的ks值
    3: IT ← 1
    4: while |V| do
    5: Vtemp ←{ }
    6: Vi.k ← $\displaystyle\sum\nolimits_{j = 1}^N { {a_{ij} } }$
    7: mindegree ← min(V.k)
    8: Vtemp ← find (V. k == mindegree)
    9: while Vtemp do
    10: Vtemp. IT←IT
    11: Vtemp.e+←$ -{\sum }_{j\in \varGamma \left(i\right)}{I}_{j}\cdot {\rm{ln}}{I}_{j}\cdot {\rm{k}}{{\rm{s}}}_{j} $
    12: endwhile
    13: delete(Vtemp)
    14: IT←IT+1
    15: VV-Vtemp
    16: endwhile
    17: ITMax ← IT
    18: while length(Rank) < N do
    19: for IT←ITMax to 1 do
    20: Vtemp = find(max(V.IT.e+))
    21: if length (Vtemp) > 1
    22: 按节点序号从大到小排序
    23: end if
    24: Rank ← { Vtemp, Rank}
    25: end for
    26: end while
    27: return Rank
    下载: 导出CSV

    表 3  由不同方法得出的排名: DC, CC, ks, Cnc, Cnc+, IKS, IE+

    Table 3.  The ranking lists determined by different methonds: DC, CC, ks, Cnc, Cnc+, IKS, IE+.

    RankDCCCksCncCnc+IKSIE+
    12111—54, 5115
    21, 4, 5, 1046—914, 571
    32, 3510—2622218
    46—9, 23213347
    511—207, 8216—896
    622, 24—2666—89, 211021
    791016510
    823923816
    916, 20, 22, 24—2615, 16, 2315234
    101710, 20, 22, 24—2629
    11others623
    122615
    1332
    1420, 22, 24, 2520, 22, 24—262
    1511—13, 14, 163
    161517
    1717—1911—14
    1818, 19
    下载: 导出CSV

    表 4  八个常见网络的基本拓扑特征, N和|E|是节点和边的数量, $ \langle d \rangle $$ \langle k \rangle $是平均距离和平均度, c是聚类系数, βthβc是流行阈值和传播值

    Table 4.  The basic topological features of the eight real neworks, N and |E|, |E| are the number of nodes and edges, $ \langle d \rangle $ and $ \langle k \rangle $are the average distance and the average degree, c is the clustering coefficient, βth and βc are the epidemic threshold and the spread value.

    NetworkN|E|$\langle d \rangle$c$ \langle k \rangle $βthβc
    NS3799146.04190.79814.82320.12470.2494
    EEC986160642.58690.450532.58420.01340.0268
    PB1222167142.73750.360027.35520.01230.0246
    Fecebook4039882343.69250.617043.69100.00940.0188
    WV70661007363.24750.209028.51290.00690.0138
    Sport13866868584.27480.276112.52810.02600.0520
    Sex15810385407.463004.87540.03650.0730
    CondMat23122934975.35230.63348.08350.04500.0900
    下载: 导出CSV

    表 5  SIR模型中节点影响指数R与五个中心性指数之间的Kendall Tau

    Table 5.  The Kendall Tau between the node influence index R of SIR model and five centrality indices.

    NetworkDCCCCncCnc+ksIKSIE+
    NS0.45930.38290.56040.70740.46430.73010.8958
    EEC0.85840.82380.89990.87710.87540.89630.9017
    PB0.84430.79560.87710.86670.86530.88590.9465
    Facebook0.62550.49480.74160.86140.67730.89260.9364
    WV0.80220.85830.89920.89390.91710.89810.9661
    Sport0.69090.68910.78750.80250.74370.85830.9197
    Sex0.41190.73290.76230.82830.51510.80650.8174
    CondMat0.59120.72680.73030.81140.64640.85650.9254
    下载: 导出CSV

    表 6  不同排序方法的单调性 M

    Table 6.  The monotonicity M of different ranking methods.

    NetworkM(DC)M(CC)M(Cnc)M(Cnc+)M(ks)M(IKS)M(IE+)
    NS0.76420.99270.93020.95930.64280.82860.9221
    EEC0.95710.98280.97480.99980.92160.93280.9881
    PB0.93280.93010.94330.95860.90630.92660.9721
    Facebook0.93980.96670.93550.96460.94190.94570.9898
    Sport0.90320.95340.92920.93770.86060.91370.9818
    Sex0.60010.91220.93320.95810.52870.92480.9989
    CondMat0.86150.95440.98710.98640.80320.90690.9996
    下载: 导出CSV
  • [1]

    Pastor-Satorras R, Vespignani A 2002 Phys. Rev. E 65 036104Google Scholar

    [2]

    Leskovec J, Adamic L A, Huberman B A 2007 Acm Trans. Web 1 5Google Scholar

    [3]

    Freeman L C 1978 Soc. Networks 1 215Google Scholar

    [4]

    Freeman L C 1977 Sociometry 40 35Google Scholar

    [5]

    Sabidussi G 1966 Psychometrika 31 581Google Scholar

    [6]

    Lü L Y, Zhou T, Zhang Q M, Stanley H E 2016 Nat. Commun. 7 10168Google Scholar

    [7]

    Lü L Y, Chen D B, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1Google Scholar

    [8]

    Brin S, Page L 1998 Comput. Netw. ISDN Syst. 30 107Google Scholar

    [9]

    Lü L Y, Zhang Y C, Yeung C H, Zhou T 2011 PloS One 6 21202Google Scholar

    [10]

    Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nat. Phys. 6 888Google Scholar

    [11]

    Pei S, Muchnik L, Andrade J S, Zheng Z M, Makse H A 2014 Sci. Rep. 4 5547Google Scholar

    [12]

    Montresor A, De Pellegrini F, Miorandi D 2011 Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing San Jose, CA, June 6–8, 2011 p207

    [13]

    Basaras P, Katsaros D, Tassiulas L 2013 Computer 46 24Google Scholar

    [14]

    Wang Z X, Zhao Y, Xi J K, Du C J 2016 Physica A 461 171Google Scholar

    [15]

    Zhou S, Mondragon R J 2004 IEEE Commun. Lett. 8 180Google Scholar

    [16]

    Wang M, Li W C, Guo Y N, Peng X Y, Li Y X 2020 Physica A 554 124229Google Scholar

    [17]

    Zareie A, Sheikhahmadi A, Jalili M, Fasaei M S K 2020 Knowledge-Based Syst. 194 105580Google Scholar

    [18]

    Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200Google Scholar

    [19]

    Hethcote H W 2000 SIAM Rev. 42 599Google Scholar

    [20]

    Ma L I, Ma C, Zhang H F, Wang B H 2016 Physica A 451 205Google Scholar

    [21]

    Li Z, Ren T, Ma X Q, Liu S M, Zhang Y X, Zhou T 2019 Sci. Rep. 9 8387Google Scholar

    [22]

    Bae J, Kim S 2014 Physica A 395 549Google Scholar

    [23]

    Bhat N, Aggarwal N, Kumar S 2020 Procedia Comput Sci. 171 662Google Scholar

    [24]

    阮逸润, 老松杨, 汤俊, 白亮 2020 物理学报 71 176401Google Scholar

    Ruan Y R, Lao S Y, Tang J, Bai L, Guo Y M 2020 Acta Phys. Sin. 71 176401Google Scholar

    [25]

    Colizza V, Flammini A, Serrano M A, Vespignani A 2006 Nat. Phys. 2 110Google Scholar

    [26]

    Rui X B, Meng F R, Wang Z X, Yuan G 2019 Appl. Intell. 49 2684Google Scholar

    [27]

    Liu D, Jing Y, Zhao J, Wang W J, Song G J 2017 Sci. Rep. 7 43330Google Scholar

    [28]

    Namtirtha A, Dutta A, Dutta B 2018 Physica A 499 310Google Scholar

    [29]

    Kim H, Anderson R 2012 Phys. Rev. E 85 026107Google Scholar

    [30]

    Takaguchi T, Sato N, Yano K, Masuda N 2012 New J. Phys. 14 093003Google Scholar

    [31]

    Qu C Q, Zhan X X, Wang G H, Wu J L, Zhang Z K 2019 Chaos 29 033116Google Scholar

    [32]

    胡钢, 许丽鹏, 徐翔 2021 物理学报 70 108901Google Scholar

    Hu G, Xu L P, Xu X 2021 Acta Phys. Sin. 70 108901Google Scholar

    [33]

    Newman M E J 2006 Phys. Rev. E 74 036104Google Scholar

    [34]

    Yin H, Benson A R, Leskovec J, Gleich D F 2017 Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (Halifax, Candana) August 13–17, 2017 p555

    [35]

    Adamic L A 2005 Glance N Proceedings of the 3rd International Workshop on Link Discovery (New York, USA) 2005 p36

    [36]

    Mcauley J, Leskovec J 2012 Proceedings of the 25th International Conference on Neural Information Processing Systems (Lake Tahoe, Nevada) 2012 p539

    [37]

    Leskovec J, Huttenlocher D, Kleinberg J 2010 Proceedings of the 19th International Conference on World Wide Web (New York, USA) 2010 p65

    [38]

    Rozemberczki B, Davies R, Sarkar R, Sutton C 2019 Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining New York, USA, 2019 p65

    [39]

    Rocha L, Liljeros F, Holme P 2011 PLoS Comput. Biol. 7 1001109Google Scholar

    [40]

    Leskovec J, Kleinberg J, Faloutsos C 2007 ACM Trans. Knowl. Discovery Data 1 2Google Scholar

    [41]

    Moreno Y, Pastor-Satorras R, Vespignani A 2002 Eur. Phys. J. B 26 521Google Scholar

    [42]

    Kenall M G 1938 Biometrika 30 81Google Scholar

    [43]

    Zhang J X, Chen D B, Dong Q, Zhao Z D 2016 Sci. Rep. 6 27823Google Scholar

    [44]

    Morone F, Makse H 2015 Nature 524 65Google Scholar

    [45]

    Goyal A, Lu W, Lakshmanan L 2011 Proceedings of the 20th International Conference on World Wide Web Hyderabad, India, 2011 p47

    [46]

    Jung K, Heo W, Chen W 2012 IEEE 12th International Conference on Data Mining Brussels, Belgium, December 10–13, 2012 p918

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  • 被引次数: 0
出版历程
  • 收稿日期:  2022-09-27
  • 修回日期:  2022-11-27
  • 上网日期:  2022-12-09
  • 刊出日期:  2023-02-20

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