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基于谱图理论的大规模复杂网络重要节点组挖掘算法

邢梓涵 刘丝语 刘慧 陈凌霄

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基于谱图理论的大规模复杂网络重要节点组挖掘算法

邢梓涵, 刘丝语, 刘慧, 陈凌霄

An algorithm for mining key node groups in large-scale complex networks based on spectral graph theory

XING Zihan, LIU Siyu, LIU Hui, CHEN Lingxiao
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  • 本文研究了无向复杂网络中基于谱图理论的节点组重要性挖掘问题. 依据复杂网络牵制控制理论中节点重要性评价指标, 删后Laplacian矩阵最小特征值较大者为重要受控节点. 本文提出一种基于多重图特征线性融合与改进贪心搜索的重要节点组挖掘方法(multi-metric fusion and enhanced greedy search algorithm, MFG算法). 该方法首先通过融合度中心性、介数中心性、K-Shell值和电阻距离等多重指标, 结合全局图特征(如图密度、平均路径长度等)构建线性加权融合模型, 预筛选候选节点组以克服单一指标的局限性; 其次, 设计二阶邻域局部扰动与全局随机游走搜索策略, 优化传统贪心算法的短视性, 在预筛选节点组中迭代选择使得删后Laplacian矩阵最小特征值最大的节点, 从而平衡局部最优与全局搜索能力; 并利用改进的反幂法进行最小特征值的计算, 降低了传统计算特征谱的复杂度, 从而使得算法总体计算性能提升. 最后, 在经典网络模型和多个真实网络中进行仿真分析, 利用不同算法挖掘重要节点组, 计算删后拉普拉斯矩阵的最小特征值, 利用SIR模型进行传播模拟, 并从网络拓扑上分析不同算法筛选出的重要节点组特征. 结果表明MFG算法相比其他几种算法挖掘重要节点组的效果更好, 对于社交网络信息传播控制具有指导意义.
    In this paper, we investigate the saliency identification of node groups in undirected complex networks by utilizing spectral graph theory of pinning control. According to the node significance criterion in network pinning control theory, where important controlled nodes are those maximizing the minimum eigenvalue of the grounded Laplacian matrix after their removal, we propose multi-metric fusion and enhanced greedy search algorithm (MFG), a novel key node group identification framework that integrates multi-metric linear fusion and an enhanced greedy search strategy. First, a linear weighted fusion model that synergistically integrates local centrality metrics with global graph properties is constructed to pre-screen potentially more important node groups, effectively reducing the inherent limitations of a single-metric evaluation paradigm. Second, a dual search strategy combining second-order neighborhood perturbation and global random walk mechanisms is developed to optimize the myopic nature of traditional greedy algorithms. Through iterative selection within pre-screened node groups, the nodes maximizing the minimum eigenvalue of the grounded Laplacian matrix are identified, achieving an optimal balance between local optimization and global search capabilities. Third, computational efficiency is enhanced by using a modified inverse power method for eigenvalue calculation, reducing the complexity of traditional spectral computations. Comprehensive simulations of generated networks and real-world networks demonstrate the framework’s superiority. The evaluation of the proposed algorithm includes three aspects: 1) comparison of the minimum eigenvalues between different algorithms; 2) SIR epidemic modeling for propagation capability assessment; 3) topological analysis of identified key nodes. The simulation results reveal the following two significant points: a) Our method outperforms state-of-the-art benchmarks (NPE, AGM, HVGC) in maximizing the ground Laplacian minimum eigenvalue in synthesized (NW small-world, ER) and real-world networks, especially at critical control sizes; b) The identified critical node groups exhibit unique topological features, typically combining high-level hubs with strategically located bridges to best balance local influence and global connectivity. Importantly, the SIR propagation model confirms that these topologically optimized populations accelerate the early outbreak of epidemics and maximize global saturation coverage, directly linking structural features with superior dynamic influence. These findings provide guidance for controlling information propagation in social networks.
  • 图 1  一个简单矩阵的电阻距离计算过程

    Fig. 1.  Process of calculating the resistance distance of a simple matrix.

    图 2  ER 随机网络节点的度及电阻距离分别与${\lambda _1}({{\boldsymbol{L}}_{N - 1}})$的排序相关性

    Fig. 2.  Correlation between the degree of nodes and the resistance distance in ER random networks, respectively, with the ${\lambda _1}({{\boldsymbol{L}}_{N - 1}})$.

    图 3  MFG算法的流程图

    Fig. 3.  Flowchart ofMFG Algorithm.

    图 4  在E-mail网络中, 取$1 \leqslant k \leqslant 12\left( {k \in \mathbb{Z}} \right)$, $s = 3$, $p = $$ 2 k$时, 使用eig函数和反幂法的程序耗时对比

    Fig. 4.  In the E-mail network, when taking $1 \leqslant k \leqslant $$ 12\left( {k \in \mathbb{Z}} \right)$, $s = 3$ and $p = 2 k$, the comparison of the computational time between the eig function and the inverse power method.

    图 5  在E-mail网络中, 分别取$k = 6, 12, 24$, $6 \leqslant p \leqslant 24$, $12 \leqslant p \leqslant 48$, $24 \leqslant p \leqslant 96$($p \in \mathbb{Z}$), $s = 3$时, 最终得到的最小特征值$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $

    Fig. 5.  In the E-mail network, when taking respectively $k = 6, 12, 24$ and $6 \leqslant p \leqslant 24$, $12 \leqslant p \leqslant 48$, $24 \leqslant p \leqslant 96$($p \in \mathbb{Z}$), $s = 3$, the final minimum eigenvalue $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $.

    图 6  在NW网络(N = 1000, Nei = 4, pc = 0.1)中, 分别取$k = 6, 12, 24$, $6 \leqslant p \leqslant 24$, $12 \leqslant p \leqslant 48$, $24 \leqslant p \leqslant 96$($p \in \mathbb{Z}$), $s = 3$时, 最终得到的最小特征值$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $

    Fig. 6.  In the NW network (N = 1000, Nei = 4, pc = 0.1), when taking respectively $k = 6, 12, 24$ and $6 \leqslant p \leqslant 24$, $12 \leqslant p \leqslant 48$, $24 \leqslant p \leqslant 96$($p \in \mathbb{Z}$), $s = 3$, the final minimum eigenvalue $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $.

    图 7  在E-mail网络中, 分别取$k = 6, 12, 24$, $p = 9, 15, 30$, 当$1 \leqslant s \leqslant 3(s \in \mathbb{Z})$, $1 \leqslant s \leqslant 6(s \in \mathbb{Z})$, $1 \leqslant s \leqslant 12(s \in \mathbb{Z})$时, 最终得到的最小特征值$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $

    Fig. 7.  In the E-mail network, when taking respectively $k = 6, 12, 24$, $p = 9, 15, 30$ and $1 \leqslant s \leqslant 3(s \in \mathbb{Z})$, $1 \leqslant s \leqslant 6(s \in \mathbb{Z})$, $1 \leqslant s \leqslant 12(s \in \mathbb{Z})$, the final minimum eigenvalue $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $.

    图 8  生成的NW小世界网络(N = 1000, Nei = 4, pc = 0.1), 标度尺反映了节点度大小的情况

    Fig. 8.  Generated NW small world network (N = 1000, Nei = 4, pc = 0.1), the scale reflects the magnitude of node degrees.

    图 9  NW网络中不同算法去除节点后不同受控节点组规模下最小特征值的比较

    Fig. 9.  Comparison of minimum eigenvalue with different target node counts following node removal by different algorithms in NW network.

    图 10  小世界网络模型中感染数随时间的变化

    Fig. 10.  Changes in the number of infections over time in the NW network model.

    图 11  生成的真实社交网络lastfm_asia (N = 7642), 标度尺反映了节点度大小的情况

    Fig. 11.  Generated real social network lastfm_asia (N = 7642), the scale reflects the magnitude of node degrees.

    图 12  lastfm_asia网络中不同算法去除节点后不同受控节点组规模下最小特征值$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $的比较

    Fig. 12.  Comparison of minimum eigenvalue $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $ with different target node counts following node removal by different algorithms in lastfm asia network.

    图 13  生成的真实社交网络E-mail (N = 1005), 标度尺反映了节点度大小的情况

    Fig. 13.  Generated real social network E-mail (N = 1005), the scale reflects the magnitude of node degrees.

    图 14  E-mail网络中不同算法去除节点后不同受控节点组规模下最小特征值$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $的比较

    Fig. 14.  Comparison of minimum eigenvalue $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $ with different target node counts following node removal by different algorithms in E-mail network.

    图 15  E-mail网络感染人数随时间的变化

    Fig. 15.  Change in the number of infections over time steps in E-mail network.

    图 16  facebook combine网络(N = 1519)

    Fig. 16.  Structure of the facebook combine Network (N = 1519).

    图 17  facebook_combine网络模型(N = 1519)中感染数随时间的变化

    Fig. 17.  Variation of the number of infections over time in the facebook_combine network model (N = 1519).

    表 1  与QR算法相比, 反幂法在乘法次数上的减少情况

    Table 1.  In comparison to the QR algorithm, the inverse power method exhibits a reduction in the number of multiplications.

    Np乘法次数减少

    100
    44.280556×108
    88.56112×108
    121.28417×109

    1000
    47.31605×1011
    81.46321×1012
    122.19481×1012

    10000
    43.73196×1015
    87.46392×1016
    121.11959×1017
    下载: 导出CSV

    表 2  不同算法在小世界网络中$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $的对比

    Table 2.  Comparison of $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $ in the NW network for different algorithms.

    受控节点组规模k度中心性算法介数中心性算法K-Shell算法NPE算法AGM
    算法
    HVGC
    算法
    MFG算法
    10.00210.00320.00090.00100.00290.00300.0032
    20.00450.00450.00110.00100.00520.00540.0059
    30.00640.00640.00120.00100.00750.00810.0086
    40.00880.00810.00160.01100.00960.01000.0113
    50.01040.00930.00170.01180.01220.01330.0141
    60.01340.01070.00330.01360.01540.01600.0169
    70.01490.01200.00350.01630.01700.01800.0194
    80.01610.01260.00360.01910.02010.02120.0223
    90.01900.01490.00370.02020.02240.02350.0249
    100.02100.01710.00380.02210.02500.02690.0273
    110.02460.01810.00390.02430.02670.02890.0299
    120.02730.01880.00410.02490.03010.03100.0324
    下载: 导出CSV

    表 3  小世界网络不同算法挖掘所得的节点重要性排序

    Table 3.  Node importance ranking by the different algorithms in NW network.

    度中心性算法介数中心性算法K-Shell算法NPE算法AGM算法HVGC
    算法
    MFG算法
    4616838616616616616
    121924839329207523523
    198329837595523207207
    236207840207371595236
    329595238924236924371
    363236239145887307595
    371382868307417417417
    417145869523329329329
    523307237339339701887
    560146240676409887701
    612814836409382382382
    616915841614937937937
    下载: 导出CSV

    表 4  不同算法在lastfm_asia网络中的$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $对比

    Table 4.  Comparison of $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $ in lastfm_asia network for different algorithms.

    受控节点组规模k度中心性算法介数中心性算法K-Shell算法NPE算法AGM算法HVGCMFG算法
    10.01480.00560.00690.00560.01480.01480.0148
    20.02890.02090.01200.02090.02890.02890.0289
    30.03700.02860.01410.02860.03960.04000.0425
    40.05090.03410.01610.03410.05080.05080.0515
    50.05680.04450.01690.03990.05720.05830.0598
    60.06090.05110.01860.04810.06250.06310.0651
    70.06260.05490.02060.05400.06410.06530.0666
    80.06510.06380.02190.06610.06500.06590.0667
    90.06600.06650.02220.06650.06610.06620.0668
    100.06680.06670.02370.06660.06680.06680.0668
    110.06680.06680.02390.06670.06690.06690.0669
    120.06680.06680.02420.06680.06690.06690.0669
    下载: 导出CSV

    表 5  lastfm_asia网络不同算法挖掘所得的节点重要性排序

    Table 5.  The node importance ranking by the different algorithms in lastfm_asia network.

    度中心性算法介数中心性算法K-Shell算法NPE算法AGM算法HVGC
    算法
    MFG算法
    723872003797200723872387238
    353172387647238610272003531
    478628559522855353161026102
    525435713354357478635314786
    34516102245354554357525525
    2511545532415128345128551796
    3598433935453451179652753451
    2855512835986102512834515275
    5128345148103545481248124812
    6102478649014901512849012855
    481235315091433952543574357
    5579310461093531285551285128
    下载: 导出CSV

    表 6  不同算法在Email网络中的$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $对比

    Table 6.  Comparison of $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $ in Email network for different algorithms.

    受控节点组规模k度中心性算法介数中心性算法K-Shell算法NPE算法AGM算法HVGC算法MFG算法
    10.648840.648840.620290.648840.648840.648840.64884
    20.652180.652550.642210.652550.652880.652880.65355
    30.653220.653500.648450.653500.654020.654020.65420
    40.653710.654020.651170.653930.654510.654510.65469
    50.654200.654290.652370.654190.654780.654780.65497
    60.654370.654460.653070.654450.655170.654730.65529
    70.654480.654590.653890.654590.655280.654870.65539
    80.654580.654700.654010.654660.655340.654980.65550
    90.654660.654760.654140.654740.655420.655050.65563
    100.654690.654820.654260.654770.655500.655240.65572
    110.654830.654860.654290.654810.655620.655300.65588
    120.654870.654870.654330.654860.655770.655460.65601
    下载: 导出CSV

    表 7  E-mail网络不同算法挖掘所得的节点重要性排序

    Table 7.  The node importance ranking by the different algorithms in E-mail network.

    度中心性算法介数中心性算法K-Shell算法NPE算法AGM算法HVGC
    算法
    MFG算法
    16116122161161161161
    12287291221228787
    836828383836
    108831151081086383
    87122129636314122
    63108130872506378
    4351416143543512214
    14378170250184108108
    1676321318413065334
    18465250167167534435
    621230413012930263
    655343726587167167
    下载: 导出CSV

    表 8  不同受控节点组规模下$ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $对比

    Table 8.  Comparison of $ {\lambda _1}\left( {{{\boldsymbol{L}}_{N - 1}}} \right) $ under different controlled node sizes.

    受控节点组规模k度中心性算法介数中心性算法K-Shell算法NPE算法AGM算法HVGC算法MFG算法
    10.01370.01370.00610.00560.01370.01370.0137
    20.15070.15070.00610.02090.15070.15070.1507
    30.22620.22620.00610.02860.22620.22620.2262
    40.22630.22620.00610..03410.36740.53150.5789
    50.22630.57890.00610.03990.58990.69870.7033
    60.22630.70280.00610.04810.76750.75430.8032
    70.22630.70350.00610.05400.82360.92751.0000
    80.22630.72880.00610.06611.00001.00001.0000
    90.22631.00000.00610.06651.00001.00001.0000
    100.22631.00000.00610.06661.00001.00001.0000
    110.22631.00000.00610.06671.00001.00001.0000
    120.22631.00000.00621.00001.00001.00001.0000
    下载: 导出CSV
  • [1]

    Liu H, Xu X H, Lu J A, Chen G R, Zeng Z G 2021 IEEE Trans. Syst. Man Cybern. Syst. 51 786Google Scholar

    [2]

    刘慧, 王炳珺, 陆君安, 李增扬 2021 物理学报 70 056401Google Scholar

    Liu H, Wang B J, Lu J A, Li Z Y 2021 Acta Phys. Sin. 70 056401Google Scholar

    [3]

    Zhou F, Su C, Xu S Q, Lü L Y 2022 Chin. Phys. B 31 068901Google Scholar

    [4]

    Dai J Y, Wang B, Sheng J F, Sun Z J, Khawaja F R, Ullah A 2019 IEEE Access 7 131719Google Scholar

    [5]

    孔江涛, 黄健, 龚建兴, 李尔玉 2018 物理学报 67 098901Google Scholar

    Kong J T, Huang J, Gong J X, Li E Y 2018 Acta Phys. Sin. 67 098901Google Scholar

    [6]

    汪亭亭, 梁宗文, 张若曦 2023 物理学报 72 048901Google Scholar

    Wang T T, Lang Z W, Zhang R X 2023 Acta Phys. Sin. 72 048901Google Scholar

    [7]

    杨松青, 蒋沅, 童天驰, 严玉为, 淦各升 2021 物理学报 70 216401Google Scholar

    Yang S Q, Jiang Y, Tong T C, Yan Y W, Gan G S 2021 Acta Phys. Sin. 70 216401Google Scholar

    [8]

    姜廷帅, 阮逸润, 李海, 白亮, 袁逸飞, 于天元 2025 物理学报 74 126401Google Scholar

    Jiang T S, Ruan Y R, Li H, Bai L, Yuan Y F, Yu T Y 2025 Acta Phys. Sin. 74 126401Google Scholar

    [9]

    Rezaei A A, Munoz J, Jalili M, Khayyam H 2023 Expert Syst. Appl. 214 119086Google Scholar

    [10]

    Zhang Y H, Lu Y L, Yang G Z, Hang Z J 2022 Appl. Sci. 12 1944Google Scholar

    [11]

    王博雅, 杨小春, 卢升荣, 唐勇平, 洪树权, 蒋惠园 2024 物理学报 73 226401Google Scholar

    Wang B Y, Yang X C, Lu S R, Tang Y P, Hong S Q, Jiang H Y 2024 Acta Phys. Sin. 73 226401Google Scholar

    [12]

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

    [13]

    Kou J H, Jia P, Liu J Y, Dai J Q, Luo H R 2023 Neurocomputing 530 23Google Scholar

    [14]

    Qiu Z H, Fan T L, Li M, Lü L Y 2021 New J. Phys. 23 033036Google Scholar

    [15]

    Chakrabarti S, Dom B, Raghavan P, Rajagopalan S, Gibson D, Kleinberg J 1998 CNIS 30 65

    [16]

    Fan C J, Zeng L, Sun Y Z, Liu Y Y 2020 Nat. Mach. Intell. 2 317Google Scholar

    [17]

    Zhao X Y, Huang B, Tang M, Zhang H F, Chen D B 2014 Europhys. Lett. 108 68005Google Scholar

    [18]

    Bao Z K, Liu J G, Zhang H F 2017 Phys. Lett. A 381 976Google Scholar

    [19]

    Ji S G, Lü L Y, Yeung C H, Hu Y Q 2017 New J. Phys. 19 073020Google Scholar

    [20]

    Anderson R M, May R M 1991 Infectious Diseases of Humans: Dynamics and Control (Oxford University Press

    [21]

    Pastor-Satorras R, Castellano C, Mieghem P V, Vespignani A 2015 Rev. Mod. Phys. 87 925Google Scholar

    [22]

    Feng M L, Zhang S F, Xia C Y, Zhao D W 2024 Chaos 34 073128Google Scholar

    [23]

    Avraam D, Hadjichrysanthou C 2024 J. Theor. Biol. 599 112010

    [24]

    Lu J A, Liu H, Chen J 2016 Synchronization in Complex Dynamical Networks (Vol. 1) (Beijing: Higher Education Press) p49

    [25]

    Pirani M, Sundaram S 2016 IEEE Trans. Autom. Control 61 509

    [26]

    Bapat R B 2010 Graphs and Matrices (Springer London

    [27]

    Zhu H Y, Klein D J, Lukovits I 1996 J. Chem. Inf. Model. 36 420

    [28]

    Gutman I, Mohar B 1996 J. Chem. Inf. Comput. Sci. 36 982Google Scholar

    [29]

    关治, 陈景良 1990 数值计算方法(北京: 清华大学出版社)

    Guan Z, Chen J L 1990 Numerical Calculation Method (Beijing: Tsinghua University Press

    [30]

    刘砚青, 陆君安 2007 复杂系统与复杂性科学 4 13

    Liu Y Q, Lu J A 2007 Complex Syst. Complexity Sci. 4 13

    [31]

    Dean J, Ghemawat S 2008 Commun. ACM 51 107

    [32]

    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 (KDD’17) Halifax, NS, Canada, August 13–17, 2017 p555

    [33]

    吴英晗, 田阔, 李明达, 胡枫 2023 计算机工程与应用 19 66

    Wu Y H, Tian K, Li M D, Hu F 2023 Comput. Appl. Eng. Educ. 19 66

    [34]

    Meng L, Xu G Q, Dong C 2025 Physica A 657 130237Google Scholar

    [35]

    Zhu S Q, Zhan J, Li X 2023 Sci. Rep. 13 16404Google Scholar

    [36]

    Snap: Stanford network analysis project https://snap.stanford.edu/.

    [37]

    Jiang W C, Wang Y H 2020 IEEE Access 8 32432Google Scholar

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  • PDF下载量:  6
  • 被引次数: 0
出版历程
  • 收稿日期:  2025-03-31
  • 修回日期:  2025-05-28
  • 上网日期:  2025-06-18

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