高阶网络统计指标综述

刘波; 曾钰洁; 杨荣湄; 吕琳媛

doi:10.7498/aps.73.20240270

摘要

复杂网络是描述和理解现实世界中复杂系统的有力工具. 近年来, 为了更准确地描述复杂网络中的交互关系, 或者从高阶视角分析成对交互作用网络, 许多学者开始使用高阶网络进行建模, 并在研究其动力学过程中发现了与成对交互作用网络不同的新现象. 然而, 与成对交互作用网络相比, 高阶网络的研究相对较少; 而且, 高阶网络结构相对复杂, 基于结构的统计指标定义较为分散且形式不统一, 这些都给描述高阶网络的拓扑结构特征带来了困难. 鉴于此, 本文综述了两种最常见的高阶网络——超图和单纯形网络——常用的统计指标及其物理意义. 本文有助于加深对高阶网络的理解, 促进对高阶网络结构特征的定量化研究, 也有助于研究者在此基础上开发更多适用于高阶网络的统计指标.

关键词:

Abstract

Complex networks serve as indispensable instruments for characterizing and understanding intricate real-world systems. Recently, researchers have delved into the realm of higher-order networks, seeking to delineate interactions within these networks with greater precision or analyze traditional pairwise networks from a higher-dimensional perspective. This effort has unearthed some new phenomena different from those observed in the traditional pairwise networks. However, despite the importance of higher-order networks, research in this area is still in its infancy. In addition, the complexity of higher-order interactions and the lack of standardized definitions for structure-based statistical indicators, also pose challenges to the investigation of higher-order networks. In recognition of these challenges, this paper presents a comprehensive survey of commonly employed statistics and their underlying physical significance in two prevalent types of higher-order networks: hypergraphs and simplicial complex networks. This paper not only outlines the specific calculation methods and application scenarios of these statistical indicators, but also provides a glimpse into future research trends. This comprehensive overview serves as a valuable resource for beginners or cross-disciplinary researchers interested in higher-order networks, enabling them to swiftly grasp the fundamental statistics pertaining to these advanced structures. By promoting a deeper understanding of higher-order networks, this paper facilitates quantitative analysis of their structural characteristics and provides guidance for researchers who aim to develop new statistical methods for higher-order networks.

Keywords:

作者及机构信息

1.
电子科技大学基础与前沿研究院, 成都　610054

2.
中国科学技术大学网络空间安全学院, 合肥　230026

3.
电子科技大学长三角研究院(湖州), 湖州　313001

通信作者: 吕琳媛, linyuan.lv@ustc.edu.cn

基金项目: 科技创新2030-“脑科学与类脑研究”重大项目青年科学家项目(批准号: 2022ZD0211400)、国家自然科学基金重大项目(批准号: T2293771)和四川省杰出青年科学基金(批准号: 2023NSFSC1919)资助的课题.

Authors and contacts

1.
Institute of Fundamental and Frontier Studies, University of Electronic Science and Technology of China, Chengdu 610054, China

2.
School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China

3.
Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China

Corresponding author: Lü Lin-Yuan, linyuan.lv@ustc.edu.cn

Funds: Project supported by the STI 2030-Major Projects (Grant No. 2022ZD0211400), the Major Program of the National Natural Science Foundation of China (Grant No. T2293771), and the Science Fund for Distinguished Young Scholars of Sichuan Provincek, China (Grant No. 2023NSFSC1919).

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参考文献

[1]	Marin A, Wellman B 2011 Social network analysis: An introduction (London: SAGE publications) pp11−25
[2]	Kossinets G, Watts D J 2006 Science 311 88 Google Scholar
[3]	Alon U 2003 Science 301 1866 Google Scholar
[4]	Alm E, Arkin A P 2003 Curr. Opin. Struct. Biol. 13 193 Google Scholar
[5]	Bose A, Clements K A 1987 Proc. IEEE 75 1607 Google Scholar
[6]	Wu F F, Varaiya P 1999 Int. J. Electr. Power Energy Syst. 21 75 Google Scholar
[7]	Williams J C, Mahmassani H S, Herman R 1987 Transp. Res. Rec. 1112 78
[8]	Verma T, Araújo N A, Herrmann H J 2014 Sci. Rep. 4 5638 Google Scholar
[9]	Strogatz S H 2001 Nature 410 268 Google Scholar
[10]	Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006 Phys. Rep. 424 175 Google Scholar
[11]	Costa L D F, Rodrigues F A, Travieso G, Villas Boas P R 2007 Adv. Phys. 56 167 Google Scholar
[12]	Barabási A L 2013 Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 371 20120375 Google Scholar
[13]	汪小帆, 李翔, 陈关荣 2012 网络科学导论 (高等教育出版社) 第82页 Wang X F, Li X, Chen G R 2012 Network Science: An Introduction (Higher Education Press) p82
[14]	周涛, 柏文洁, 汪秉宏, 刘之景, 严钢 2005 物理 34 31 Google Scholar Zhou T, Bai W J, Wang B H, Liu Z J, Yan G 2005 Physics 34 31 Google Scholar
[15]	Courtney O T, Bianconi G 2017 Phys. Rev. E 95 062301 Google Scholar
[16]	Lung R I, Gaskó N, Suciu M A 2018 Scientometrics 117 1361 Google Scholar
[17]	Pearcy N, Crofts J J, Chuzhanova N 2014 Int. J. Biol. Vet. Agric. Food Eng. 8 752
[18]	Mastrandrea R, Fournet J, Barrat A 2015 PloS One 10 e0136497 Google Scholar
[19]	Stehlé J, Voirin N, Barrat A, et al. 2011 PloS One 6 e23176 Google Scholar
[20]	Battiston F, Cencetti G, Iacopini I, Latora V, Lucas M, Patania A, Young J G, Petri G 2020 Phys. Rep. 874 1 Google Scholar
[21]	Battiston F, Amico E, Barrat A, et al. 2021 Nat. Phys. 17 1093 Google Scholar
[22]	Bianconi G 2021 Higher-order Networks (Cambridge: Cambridge University Press) pp7–45
[23]	Shi D, Chen G 2022 Natl. Sci. Rev. 9 nwac038 Google Scholar
[24]	Zhao D, Li R, Peng H, Zhong M, Wang W 2022 Chaos Solit. Fractals 155 111701 Google Scholar
[25]	Wang W, Li W, Lin T, Wu T, Pan L, Liu Y 2022 Appl. Math. Comput. 420 126793 Google Scholar
[26]	Millán A P, Torres J J, Bianconi G 2020 Phys. Rev. Lett. 124 218301 Google Scholar
[27]	Lucas M, Cencetti G, Battiston F 2020 Phys. Rev. Res. 2 033410 Google Scholar
[28]	Iacopini I, Petri G, Barrat A, Latora V 2019 Nat. Commun. 10 1 Google Scholar
[29]	Chowdhary S, Kumar A, Cencetti G, Iacopini I, Battiston F 2021 J. Phys.: Complex. 2 035019 Google Scholar
[30]	陈浩宇, 徐涛, 刘闯, 张子柯, 詹秀秀 2024 物理学报 73 038901 Google Scholar Chen H Y, Xu T, Liu C, Zhang Z K, Zhan X X 2024 Acta Phys. Sin. 73 038901 Google Scholar
[31]	Gómez-Gardenes J, Gómez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701 Google Scholar
[32]	Kovalenko K, Dai X, Alfaro-Bittner K, Raigorodskii A, Perc M, Boccaletti S 2021 Phys. Rev. Lett. 127 258301 Google Scholar
[33]	Tanaka T, Aoyagi T 2011 Phys. Rev. Lett. 106 224101 Google Scholar
[34]	Zhang Y, Latora V, Motter A E 2021 Commun. Phys. 4 195 Google Scholar
[35]	Kundu S, Ghosh D 2022 Phys. Rev. E 105 L042202 Google Scholar
[36]	Bick C, Ashwin P, Rodrigues A 2016 Chaos 26 094814 Google Scholar
[37]	Wang W, Wang Z X, Cai S M 2018 Phys. Rev. E 98 052312 Google Scholar
[38]	Guilbeault D, Becker J, Centola D 2018 Complex Spreading Phenomena in Social Systems (Cham: Springer) pp3−25
[39]	Wang W, Liu Q H, Liang J, Hu Y, Zhou T 2019 Phys. Rep. 820 1 Google Scholar
[40]	Wang D, Zhao Y, Luo J, Leng H 2021 Chaos: Interdiscip. J. Nonlinear Sci. 31 053112 Google Scholar
[41]	王兆慧, 沈华伟, 曹婍, 程学旗 2011 软件学报 33 171 Google Scholar Wang Z H, Shen H W, Cao Q, Cheng X Q 2011 J. Softw. 33 171 Google Scholar
[42]	Lü L, Chen D, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1 Google Scholar
[43]	任晓龙, 吕琳媛 2014 科学通报 59 1175 Google Scholar Ren X L, Lü L Y 2014 Chin. Sci. Bull. 59 1175 Google Scholar
[44]	李江, 刘影, 王伟, 周涛 2024 物理学报 73 048901 Google Scholar Li J, Liu Y, Wang W, Zhou T 2024 Acta Phys. Sin. 73 048901 Google Scholar
[45]	Lü L, Zhou T 2011 Phys. A: Stat. Mech. Appl. 390 1150 Google Scholar
[46]	Liu B, Yang R, Lü L 2023 Chaos: Interdiscip. J. Nonlinear Sci. 33 083108 Google Scholar
[47]	吕琳媛 2010 电子科技大学学报 39 651 Google Scholar Lü L Y 2010 J. Univ. Electron. Sci. Technol. China 39 651 Google Scholar
[48]	Newman M E 2006 Proc. Natl. Acad. Sci. 103 8577 Google Scholar
[49]	Jiang Y, Jia C, Yu J 2013 Phys. A: Stat. Mech. Appl. 392 2182 Google Scholar
[50]	Watts D J, Strogatz S H 1998 Nature 393 440 Google Scholar
[51]	Barabási A L, Albert R 1999 Science 286 509 Google Scholar
[52]	许小可, 崔文阔, 崔丽艳, 肖婧, 尚可可 2019 电子科技大学学报 48 122 Google Scholar Xu X K, Cui W K, Cui L Y, Xiao J, Shang K K 2019 J. Univ. Electron. Sci. Technol. China 48 122 Google Scholar
[53]	Zeng Y, Liu B, Zhou F, Lü L 2023 Entropy 25 1390 Google Scholar
[54]	Bick C, Gross E, Harrington H A, Schaub M T 2023 SIAM Rev. 65 686 Google Scholar
[55]	Feng Y, You H, Zhang Z, Ji R, Gao Y 2019 Proceedings of the AAAI Conference on Artificial Intelligence 33 3558 Google Scholar
[56]	Zhu J, Zhu J, Ghosh S, Wu W, Yuan J 2018 IEEE Trans. Netw. Sci. Eng. 6 801 Google Scholar
[57]	Viñas R, Joshi C K, Georgiev D, Lin P, Dumitrascu B, Gamazon E R, Liò P 2023 Nat. Mach. Intell. 5 739 Google Scholar
[58]	Huang J, Zhang S, Yang F, Yu T, Prasad L N, Guduri M, Yu K 2023 IEEE Trans. Consum. Electron. 1 1775 Google Scholar
[59]	Ruggeri N, Contisciani M, Battiston F, De Bacco C 2023 Sci. Adv. 9 eadg9159 Google Scholar
[60]	Wu H, Li N, Zhang J, Chen S, Ng M K, Long J 2024 Pattern Recognit. 146 109995 Google Scholar
[61]	Mancastroppa M, Iacopini I, Petri G, Barrat A 2023 Nat. Commun. 14 6223 Google Scholar
[62]	Gao Y, Feng Y, Ji S, Ji R 2022 IEEE Trans. Pattern Anal. Mach. Intell. 45 3181 Google Scholar
[63]	Li Z, Deng Z, Han Z, Alfaro-Bittner K, Barzel B, Boccaletti S 2021 Chaos Solit. Fractals 152 111307 Google Scholar
[64]	Gambuzza L V, Di Patti F, Gallo L, et al. 2021 Nat. Commun. 12 1255 Google Scholar
[65]	Wang H, Ma C, Chen H S, Lai Y C, Zhang H F 2022 Nat. Commun. 13 3043 Google Scholar
[66]	Benson A R, Abebe R, Schaub M T, Jadbabaie A, Kleinberg J 2018 Proc. Natl. Acad. Sci. 115 E11221 Google Scholar
[67]	Shi D, Chen Z, Sun X, Chen Q, Ma C, Lou Y, Chen G 2021 Commun. Phys. 4 249 Google Scholar
[68]	Reimann M W, Nolte M, Scolamiero M, et al. 2017 Front. Comput. Neurosci. 11 48 Google Scholar
[69]	Sizemore A E, Giusti C, Kahn A, Vettel J M, Betzel R F, Bassett D S 2018 J. Comput. Neurosci. 44 115 Google Scholar
[70]	Kovalenko K, Sendiña-Nadal I, Khalil N, et al. 2021 Commun. Phys. 4 43 Google Scholar
[71]	Holland P W, Leinhardt S 1971 Comp. Group Stud. 2 107 Google Scholar
[72]	Estrada E, Rodríguez-Velázquez J A 2006 Phys. A: Stat. Mech. Appl. 364 581 Google Scholar
[73]	Carletti T, Battiston F, Cencetti G, Fanelli D 2020 Phys. Rev. E 101 022308 Google Scholar
[74]	Carletti T, Fanelli D, Lambiotte R 2021 J. Phys.: Complex. 2 015011 Google Scholar
[75]	Aksoy S G, Joslyn C, Marrero C O, Praggastis B, Purvine E 2020 EPJ Data Sci. 9 16 Google Scholar
[76]	Lu L, Peng X 2013 Internet Math. 9 3 Google Scholar
[77]	Vasilyeva E, Romance M, Samoylenko I, Kovalenko K, Musatov D, Raigorodskii A M, Boccaletti S 2023 Entropy 25 923 Google Scholar
[78]	Gao J, Zhao Q, Ren W, Swami A, Ramanathan R, Bar-Noy A 2014 IEEE/ACM Trans. Netw. 23 1805 Google Scholar
[79]	Zlatić V, Ghoshal G, Caldarelli G 2009 Phys. Rev. E 80 036118 Google Scholar
[80]	Bauer F, Hua B, Jost J, Liu S, Wang G 2017 Modern Approaches to Discrete Curvature (Cham: Springer) pp1−62
[81]	Samal A, Sreejith R, Gu J, Liu S, Saucan E, Jost J 2018 Sci. Rep. 8 8650 Google Scholar
[82]	Leal W, Restrepo G, Stadler P F, Jost J 2021 Adv. Complex Syst. 24 2150003 Google Scholar
[83]	Eidi M, Farzam A, Leal W, Samal A, Jost J 2020 Theory Biosci. 139 337 Google Scholar
[84]	Bauer F, J Jost, S Liu 2012 Math. Res. Lett. 19 1185
[85]	Eidi M, Jost J 2020 Sci. Rep. 10 12466 Google Scholar
[86]	Kapoor K, Sharma D, Srivastava J 2013 IEEE 2nd Network Science Workshop New York, USA, April 29–May 1, 2013 p152
[87]	Granovetter M S 1973 Am. J. Sociol. 78 1360 Google Scholar
[88]	Dorogovtsev S N, Goltsev A V, Mendes J F F 2006 Phys. Rev. Lett. 96 040601 Google Scholar
[89]	Xiao Q 2013 Res. J. Appl. Sci. Eng. Technol. 5 568 Google Scholar
[90]	Lee J, Lee Y, Oh S M, Kahng B 2021 Chaos: Interdiscip. J. Nonlinear Sci. 31 061108 Google Scholar
[91]	Bonacich P 1972 J. Math. Sociol. 2 113 Google Scholar
[92]	Benson A R 2019 SIAM J. Math. Data Sci. 1 293 Google Scholar
[93]	Lemmens B, Nussbaum R 2012 Nonlinear Perron-frobenius Theory (Vol. 189) (Cambridge: Cambridge University Press) pp2−4
[94]	Clausius R 1879 The Mechanical Theory of Heat (Macmillan) pp327−365
[95]	Shannon C E 1948 Bell Syst. Tech. J. 27 379 Google Scholar
[96]	Bloch I, Bretto A 2019 Discrete Geometry for Computer Imagery: 21st IAPR International Conference Marne-la-Vallée, France, March 26–28, 2019 pp143–154
[97]	Hu D, Li X L, Liu X G, Zhang S G 2019 Acta Math. Sin. Engl. Ser. 35 1238 Google Scholar
[98]	Wang H, Xiao G, Yan Y, Suter D 2018 IEEE Trans. Pattern Anal. Mach. Intell. 41 697 Google Scholar
[99]	Goldberg T E 2002 Sr. Thesis Bard Coll. 6 25
[100]	Estrada E, Ross G J 2018 J. Theor. Biol. 438 46 Google Scholar
[101]	Serrano D H, Hernández-Serrano J, Gómez D S 2020 Chaos Solit. Fractals 137 109839 Google Scholar
[102]	Serrano D H, Gómez D S 2020 Appl. Math. Comput. 382 125331 Google Scholar
[103]	Bonacich P 2007 Soc. Netw. 29 555 Google Scholar
[104]	Katz L 1953 Psychometrika 18 39 Google Scholar
[105]	Estrada E, Knight P A 2015 A First Course in Network Theory (Oxford: Oxford University Press) pp157−160
[106]	Okamoto K, Chen W, Li X Y 2008 International Workshop on Frontiers in Algorithmics Changsha China, June 19–21, 2008 pp186–195
[107]	Newman M E 2005 Soc. Netw. 27 39 Google Scholar
[108]	Maletić S, Rajković M, Vasiljević D 2008 Computational Science–ICCS 2008: 8th International Conference Kraków Poland, June 23–25, 2008 pp568–575
[109]	Shi D, Lü L, Chen G 2019 Natl. Sci. Rev. 6 962 Google Scholar

施引文献

图 1 两种不同类型超图的简单示例　(a) 一个拥有11个节点的超图; (b) 一个拥有9个节点的3-均匀超图

Fig. 1. A simple example of two different types of hypergraphs: (a) Simple hypergraph with 11 nodes; (b) 3-uniform hypergraph with 9 nodes.

下载: 全尺寸图片幻灯片

图 2 单纯形网络相关示意图　(a) 一组时序高阶交互数据; (b) 一个11节点的单纯形网络; (c) 基于图(b)中单纯形网络的骨架网络; (d)一个11节点的团复形网络

Fig. 2. Correlation diagrams of the simplicial network: (a) A set of temporal higher-order interaction data; (b) a simplicial network with 11 nodes; (c) a skeleton network based on the simplicial network in Fig. 2(b); (d) a clique complex with 11 nodes.

下载: 全尺寸图片幻灯片

图 3 一个3条超边的超图和其对应的加权线图^[73]　(a) 一个3条超边的超图; (b) 图3(a)对应的加权线图

Fig. 3. A hypergraph with 3 hyperedges and its corresponding weighted line graph^[73]: (a) A hypergraph with 3 hyperedges; (b) a weighted line graph corresponding to Fig. 3(a).

下载: 全尺寸图片幻灯片

图 4 不同情形下两个超节点之间的路径示意图^[73]

Fig. 4. Diagram of the path between two hypernodes in different cases^[73].

下载: 全尺寸图片幻灯片

图 5 一个具有5条超边的超图的k-核分解示意图^[61]

Fig. 5. Diagram of a k-core decomposition of a hypergraph with 5 hyperedges^[61].

下载: 全尺寸图片幻灯片

表 1 基于超图的统计指标总结

Table 1. Summary of statistical indicators of the hypergraph

指标类型	指标名称
度相关指标	度、超度、超边度、余平均度
聚集系数	节点的聚集系数、网络的聚集系数
距离相关指标	路径长度、超节点之间的距离
密度相关指标	超边密度、超图密度
曲率相关指标	Forman-Ricci曲率、Ollivier-Ricci曲率
中心性指标	度中心性、核心度中心性、接近中心性、介数中心性、特征向量中心性
熵相关指标	超图熵、超图的香农熵、加权超图的超图熵

下载: 导出CSV

表 2 基于单纯形网络的统计指标总结

Table 2. Summary of statistical indicators of the simplicial network

指标类型	指标名称
度相关指标	上邻接度、下邻接度、度、上 p 邻接度、下 p 邻接度、严格上 p 邻接度、严格下 p 邻接度、上$(h, p)$邻接度、下$(h, p)$邻接度、p 邻接度、最大 p 邻接度、最大单纯形度
路径和距离相关指标	$s_k$游走、p 游走、最短路径长度、离心率、直径
中心性指标	度中心性、特征向量中心性、Katz中心性、接近中心性、介数中心性
聚集系数	聚集系数
拓扑不变量	贝蒂数、欧拉示性数

下载: 导出CSV

[1]	Marin A, Wellman B 2011 Social network analysis: An introduction (London: SAGE publications) pp11−25
[2]	Kossinets G, Watts D J 2006 Science 311 88 Google Scholar
[3]	Alon U 2003 Science 301 1866 Google Scholar
[4]	Alm E, Arkin A P 2003 Curr. Opin. Struct. Biol. 13 193 Google Scholar
[5]	Bose A, Clements K A 1987 Proc. IEEE 75 1607 Google Scholar
[6]	Wu F F, Varaiya P 1999 Int. J. Electr. Power Energy Syst. 21 75 Google Scholar
[7]	Williams J C, Mahmassani H S, Herman R 1987 Transp. Res. Rec. 1112 78
[8]	Verma T, Araújo N A, Herrmann H J 2014 Sci. Rep. 4 5638 Google Scholar
[9]	Strogatz S H 2001 Nature 410 268 Google Scholar
[10]	Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006 Phys. Rep. 424 175 Google Scholar
[11]	Costa L D F, Rodrigues F A, Travieso G, Villas Boas P R 2007 Adv. Phys. 56 167 Google Scholar
[12]	Barabási A L 2013 Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 371 20120375 Google Scholar
[13]	汪小帆, 李翔, 陈关荣 2012 网络科学导论 (高等教育出版社) 第82页 Wang X F, Li X, Chen G R 2012 Network Science: An Introduction (Higher Education Press) p82
[14]	周涛, 柏文洁, 汪秉宏, 刘之景, 严钢 2005 物理 34 31 Google Scholar Zhou T, Bai W J, Wang B H, Liu Z J, Yan G 2005 Physics 34 31 Google Scholar
[15]	Courtney O T, Bianconi G 2017 Phys. Rev. E 95 062301 Google Scholar
[16]	Lung R I, Gaskó N, Suciu M A 2018 Scientometrics 117 1361 Google Scholar
[17]	Pearcy N, Crofts J J, Chuzhanova N 2014 Int. J. Biol. Vet. Agric. Food Eng. 8 752
[18]	Mastrandrea R, Fournet J, Barrat A 2015 PloS One 10 e0136497 Google Scholar
[19]	Stehlé J, Voirin N, Barrat A, et al. 2011 PloS One 6 e23176 Google Scholar
[20]	Battiston F, Cencetti G, Iacopini I, Latora V, Lucas M, Patania A, Young J G, Petri G 2020 Phys. Rep. 874 1 Google Scholar
[21]	Battiston F, Amico E, Barrat A, et al. 2021 Nat. Phys. 17 1093 Google Scholar
[22]	Bianconi G 2021 Higher-order Networks (Cambridge: Cambridge University Press) pp7–45
[23]	Shi D, Chen G 2022 Natl. Sci. Rev. 9 nwac038 Google Scholar
[24]	Zhao D, Li R, Peng H, Zhong M, Wang W 2022 Chaos Solit. Fractals 155 111701 Google Scholar
[25]	Wang W, Li W, Lin T, Wu T, Pan L, Liu Y 2022 Appl. Math. Comput. 420 126793 Google Scholar
[26]	Millán A P, Torres J J, Bianconi G 2020 Phys. Rev. Lett. 124 218301 Google Scholar
[27]	Lucas M, Cencetti G, Battiston F 2020 Phys. Rev. Res. 2 033410 Google Scholar
[28]	Iacopini I, Petri G, Barrat A, Latora V 2019 Nat. Commun. 10 1 Google Scholar
[29]	Chowdhary S, Kumar A, Cencetti G, Iacopini I, Battiston F 2021 J. Phys.: Complex. 2 035019 Google Scholar
[30]	陈浩宇, 徐涛, 刘闯, 张子柯, 詹秀秀 2024 物理学报 73 038901 Google Scholar Chen H Y, Xu T, Liu C, Zhang Z K, Zhan X X 2024 Acta Phys. Sin. 73 038901 Google Scholar
[31]	Gómez-Gardenes J, Gómez S, Arenas A, Moreno Y 2011 Phys. Rev. Lett. 106 128701 Google Scholar
[32]	Kovalenko K, Dai X, Alfaro-Bittner K, Raigorodskii A, Perc M, Boccaletti S 2021 Phys. Rev. Lett. 127 258301 Google Scholar
[33]	Tanaka T, Aoyagi T 2011 Phys. Rev. Lett. 106 224101 Google Scholar
[34]	Zhang Y, Latora V, Motter A E 2021 Commun. Phys. 4 195 Google Scholar
[35]	Kundu S, Ghosh D 2022 Phys. Rev. E 105 L042202 Google Scholar
[36]	Bick C, Ashwin P, Rodrigues A 2016 Chaos 26 094814 Google Scholar
[37]	Wang W, Wang Z X, Cai S M 2018 Phys. Rev. E 98 052312 Google Scholar
[38]	Guilbeault D, Becker J, Centola D 2018 Complex Spreading Phenomena in Social Systems (Cham: Springer) pp3−25
[39]	Wang W, Liu Q H, Liang J, Hu Y, Zhou T 2019 Phys. Rep. 820 1 Google Scholar
[40]	Wang D, Zhao Y, Luo J, Leng H 2021 Chaos: Interdiscip. J. Nonlinear Sci. 31 053112 Google Scholar
[41]	王兆慧, 沈华伟, 曹婍, 程学旗 2011 软件学报 33 171 Google Scholar Wang Z H, Shen H W, Cao Q, Cheng X Q 2011 J. Softw. 33 171 Google Scholar
[42]	Lü L, Chen D, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1 Google Scholar
[43]	任晓龙, 吕琳媛 2014 科学通报 59 1175 Google Scholar Ren X L, Lü L Y 2014 Chin. Sci. Bull. 59 1175 Google Scholar
[44]	李江, 刘影, 王伟, 周涛 2024 物理学报 73 048901 Google Scholar Li J, Liu Y, Wang W, Zhou T 2024 Acta Phys. Sin. 73 048901 Google Scholar
[45]	Lü L, Zhou T 2011 Phys. A: Stat. Mech. Appl. 390 1150 Google Scholar
[46]	Liu B, Yang R, Lü L 2023 Chaos: Interdiscip. J. Nonlinear Sci. 33 083108 Google Scholar
[47]	吕琳媛 2010 电子科技大学学报 39 651 Google Scholar Lü L Y 2010 J. Univ. Electron. Sci. Technol. China 39 651 Google Scholar
[48]	Newman M E 2006 Proc. Natl. Acad. Sci. 103 8577 Google Scholar
[49]	Jiang Y, Jia C, Yu J 2013 Phys. A: Stat. Mech. Appl. 392 2182 Google Scholar
[50]	Watts D J, Strogatz S H 1998 Nature 393 440 Google Scholar
[51]	Barabási A L, Albert R 1999 Science 286 509 Google Scholar
[52]	许小可, 崔文阔, 崔丽艳, 肖婧, 尚可可 2019 电子科技大学学报 48 122 Google Scholar Xu X K, Cui W K, Cui L Y, Xiao J, Shang K K 2019 J. Univ. Electron. Sci. Technol. China 48 122 Google Scholar
[53]	Zeng Y, Liu B, Zhou F, Lü L 2023 Entropy 25 1390 Google Scholar
[54]	Bick C, Gross E, Harrington H A, Schaub M T 2023 SIAM Rev. 65 686 Google Scholar
[55]	Feng Y, You H, Zhang Z, Ji R, Gao Y 2019 Proceedings of the AAAI Conference on Artificial Intelligence 33 3558 Google Scholar
[56]	Zhu J, Zhu J, Ghosh S, Wu W, Yuan J 2018 IEEE Trans. Netw. Sci. Eng. 6 801 Google Scholar
[57]	Viñas R, Joshi C K, Georgiev D, Lin P, Dumitrascu B, Gamazon E R, Liò P 2023 Nat. Mach. Intell. 5 739 Google Scholar
[58]	Huang J, Zhang S, Yang F, Yu T, Prasad L N, Guduri M, Yu K 2023 IEEE Trans. Consum. Electron. 1 1775 Google Scholar
[59]	Ruggeri N, Contisciani M, Battiston F, De Bacco C 2023 Sci. Adv. 9 eadg9159 Google Scholar
[60]	Wu H, Li N, Zhang J, Chen S, Ng M K, Long J 2024 Pattern Recognit. 146 109995 Google Scholar
[61]	Mancastroppa M, Iacopini I, Petri G, Barrat A 2023 Nat. Commun. 14 6223 Google Scholar
[62]	Gao Y, Feng Y, Ji S, Ji R 2022 IEEE Trans. Pattern Anal. Mach. Intell. 45 3181 Google Scholar
[63]	Li Z, Deng Z, Han Z, Alfaro-Bittner K, Barzel B, Boccaletti S 2021 Chaos Solit. Fractals 152 111307 Google Scholar
[64]	Gambuzza L V, Di Patti F, Gallo L, et al. 2021 Nat. Commun. 12 1255 Google Scholar
[65]	Wang H, Ma C, Chen H S, Lai Y C, Zhang H F 2022 Nat. Commun. 13 3043 Google Scholar
[66]	Benson A R, Abebe R, Schaub M T, Jadbabaie A, Kleinberg J 2018 Proc. Natl. Acad. Sci. 115 E11221 Google Scholar
[67]	Shi D, Chen Z, Sun X, Chen Q, Ma C, Lou Y, Chen G 2021 Commun. Phys. 4 249 Google Scholar
[68]	Reimann M W, Nolte M, Scolamiero M, et al. 2017 Front. Comput. Neurosci. 11 48 Google Scholar
[69]	Sizemore A E, Giusti C, Kahn A, Vettel J M, Betzel R F, Bassett D S 2018 J. Comput. Neurosci. 44 115 Google Scholar
[70]	Kovalenko K, Sendiña-Nadal I, Khalil N, et al. 2021 Commun. Phys. 4 43 Google Scholar
[71]	Holland P W, Leinhardt S 1971 Comp. Group Stud. 2 107 Google Scholar
[72]	Estrada E, Rodríguez-Velázquez J A 2006 Phys. A: Stat. Mech. Appl. 364 581 Google Scholar
[73]	Carletti T, Battiston F, Cencetti G, Fanelli D 2020 Phys. Rev. E 101 022308 Google Scholar
[74]	Carletti T, Fanelli D, Lambiotte R 2021 J. Phys.: Complex. 2 015011 Google Scholar
[75]	Aksoy S G, Joslyn C, Marrero C O, Praggastis B, Purvine E 2020 EPJ Data Sci. 9 16 Google Scholar
[76]	Lu L, Peng X 2013 Internet Math. 9 3 Google Scholar
[77]	Vasilyeva E, Romance M, Samoylenko I, Kovalenko K, Musatov D, Raigorodskii A M, Boccaletti S 2023 Entropy 25 923 Google Scholar
[78]	Gao J, Zhao Q, Ren W, Swami A, Ramanathan R, Bar-Noy A 2014 IEEE/ACM Trans. Netw. 23 1805 Google Scholar
[79]	Zlatić V, Ghoshal G, Caldarelli G 2009 Phys. Rev. E 80 036118 Google Scholar
[80]	Bauer F, Hua B, Jost J, Liu S, Wang G 2017 Modern Approaches to Discrete Curvature (Cham: Springer) pp1−62
[81]	Samal A, Sreejith R, Gu J, Liu S, Saucan E, Jost J 2018 Sci. Rep. 8 8650 Google Scholar
[82]	Leal W, Restrepo G, Stadler P F, Jost J 2021 Adv. Complex Syst. 24 2150003 Google Scholar
[83]	Eidi M, Farzam A, Leal W, Samal A, Jost J 2020 Theory Biosci. 139 337 Google Scholar
[84]	Bauer F, J Jost, S Liu 2012 Math. Res. Lett. 19 1185
[85]	Eidi M, Jost J 2020 Sci. Rep. 10 12466 Google Scholar
[86]	Kapoor K, Sharma D, Srivastava J 2013 IEEE 2nd Network Science Workshop New York, USA, April 29–May 1, 2013 p152
[87]	Granovetter M S 1973 Am. J. Sociol. 78 1360 Google Scholar
[88]	Dorogovtsev S N, Goltsev A V, Mendes J F F 2006 Phys. Rev. Lett. 96 040601 Google Scholar
[89]	Xiao Q 2013 Res. J. Appl. Sci. Eng. Technol. 5 568 Google Scholar
[90]	Lee J, Lee Y, Oh S M, Kahng B 2021 Chaos: Interdiscip. J. Nonlinear Sci. 31 061108 Google Scholar
[91]	Bonacich P 1972 J. Math. Sociol. 2 113 Google Scholar
[92]	Benson A R 2019 SIAM J. Math. Data Sci. 1 293 Google Scholar
[93]	Lemmens B, Nussbaum R 2012 Nonlinear Perron-frobenius Theory (Vol. 189) (Cambridge: Cambridge University Press) pp2−4
[94]	Clausius R 1879 The Mechanical Theory of Heat (Macmillan) pp327−365
[95]	Shannon C E 1948 Bell Syst. Tech. J. 27 379 Google Scholar
[96]	Bloch I, Bretto A 2019 Discrete Geometry for Computer Imagery: 21st IAPR International Conference Marne-la-Vallée, France, March 26–28, 2019 pp143–154
[97]	Hu D, Li X L, Liu X G, Zhang S G 2019 Acta Math. Sin. Engl. Ser. 35 1238 Google Scholar
[98]	Wang H, Xiao G, Yan Y, Suter D 2018 IEEE Trans. Pattern Anal. Mach. Intell. 41 697 Google Scholar
[99]	Goldberg T E 2002 Sr. Thesis Bard Coll. 6 25
[100]	Estrada E, Ross G J 2018 J. Theor. Biol. 438 46 Google Scholar
[101]	Serrano D H, Hernández-Serrano J, Gómez D S 2020 Chaos Solit. Fractals 137 109839 Google Scholar
[102]	Serrano D H, Gómez D S 2020 Appl. Math. Comput. 382 125331 Google Scholar
[103]	Bonacich P 2007 Soc. Netw. 29 555 Google Scholar
[104]	Katz L 1953 Psychometrika 18 39 Google Scholar
[105]	Estrada E, Knight P A 2015 A First Course in Network Theory (Oxford: Oxford University Press) pp157−160
[106]	Okamoto K, Chen W, Li X Y 2008 International Workshop on Frontiers in Algorithmics Changsha China, June 19–21, 2008 pp186–195
[107]	Newman M E 2005 Soc. Netw. 27 39 Google Scholar
[108]	Maletić S, Rajković M, Vasiljević D 2008 Computational Science–ICCS 2008: 8th International Conference Kraków Poland, June 23–25, 2008 pp568–575
[109]	Shi D, Lü L, Chen G 2019 Natl. Sci. Rev. 6 962 Google Scholar

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