Identifying influential nodes in spreading process  in higher-order networks

Li Jiang; Liu Ying; Wang Wei; Zhou Tao

doi:10.7498/aps.73.20231416

Abstract

Identifying influential nodes in spreading process in the network is an important step to control the speed and range of spreading, which can be used to accelerate the spread of beneficial information such as healthy behaviors, innovations and suppress the spread of epidemics, rumors and fake news. Existing researches on identification of influential spreaders are mostly based on low-order complex networks with pairwise interactions. However, interactions between individuals occur not only between pairwise nodes but also in groups of three or more nodes, which introduces complex mechanism of reinforcement and indirect influence. The higher-order networks such as simplicial complexes and hypergraphs, can describe features of interactions that go beyond the limitation of pairwise interactions. Currently, there are relatively few researches of identifying influential spreaders in higher-order networks. Some centralities of nodes such as higher-order degree centrality and eigenvector centrality are proposed, but they mostly consider only the network structure. As for identification of influential spreaders, the spreading influence of a node is closely related to the spreading process. In this paper, we work on identification of influential spreaders on simplicial complexes by taking both network structure and dynamical process into consideration. Firstly, we quantitatively describe the dynamics of disease spreading on simplicial complexes by using the Susceptible-Infected-Recovered microscopic Markov equations. Next, we use the microscopic Markov equations to calculate the probability that a node is infected in the spreading process, which is defined as the spreading centrality (SC) of nodes. This spreading centrality involves both the structure of simplicial complex and the dynamical process on it, and is then used to rank the spreading influence of nodes. Simulation results on two types of synthetic simplicial complexes and four real simplicial complexes show that compared with the existing centralities on higher-order networks and the optimal centralities of collective influence and nonbacktracking centrality in complex networks, the proposed spreading centrality can more accurately identify the most influential spreaders in simplicial complexes. In addition, we find that the probability of nodes infected is highly positively correlated with its influence, which is because disease preferentially reaches nodes with many contacts, who can in turn infect their many neighbors and become influential spreaders.

Keywords:

Authors and contacts

1.
School of Computer Science, Southwest Petroleum University, Chengdu 610500, China
2.
Department of Physics, University of Fribourg, Fribourg 1700, Switzerland
3.
College of Public Health, Chongqing Medical University, Chongqing 400016, China
4.
Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China

Corresponding author: Liu Ying, shinningliu@163.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61802321, 61903266), the Science and Technology Program of Sichuan Province, China (Grant No. 2023YFG0129), and the China Postdoctoral Science Fundation-Special Fund (Grant No. 2019T120829).

FullText HTML : translate this paragraph

References

[1]	Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200 Google Scholar
[2]	Moreno Y, Nekovee M, Pacheco A F 2004 Phys. Rev. E 69 066130 Google Scholar
[3]	Motter A E 2004 Phys. Rev. Lett. 93 098701 Google Scholar
[4]	Li D, Fu B, Wang Y, Lu G, Berezin Y, Stanley H E, Havlin S 2014 Proc. Natl. Acad. Sci. 112 669
[5]	Kephart J O, Sorkin G B, Chess D M, White S R 1997 Sci. Am. 277 88
[6]	Hale T, Angrist N, Goldszmidt R, et al. 2023 Nat. Hum. Behav. 5 529 Google Scholar
[7]	Rocha Y M, de Moura G A, Desidério G A, et al. 2023 J. Public Health 31 1007 Google Scholar
[8]	Schäfer B, Witthaut D, Timme M, Latora V 2018 Nat. Commun. 9 1975 Google Scholar
[9]	任晓龙, 吕琳媛 2014 科学通报 59 1175 Google Scholar Ren X L, Lü L Y 2014 Sci. Bull. 59 1175 Google Scholar
[10]	Yang K C, Pierri F, Hui P M, Axelrod D, Torres-Lugo C, Bryden J, Menczer F 2021 Big Data Soc. 8 1
[11]	Nielsen B F, Simonsen L, Sneppen K 2021 Phys. Rev. Lett. 126 118301 Google Scholar
[12]	Freeman L C 1978 Soc. Networks 1 215 Google Scholar
[13]	Lü L, Zhou T, Zhang Q M, Stanley H E 2016 Nat. Commun. 7 10168 Google Scholar
[14]	Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nat. Phys. 6 888 Google Scholar
[15]	Morone F, Makse H A 2015 Nature 524 65 Google Scholar
[16]	Sabidussi G 1966 Psychometrika 31 581 Google Scholar
[17]	Freeman L C 1977 Sociometry 40 35 Google Scholar
[18]	Estrada E, Rodríguez-Velázquez J A 2005 Phys. Rev. E 71 056103 Google Scholar
[19]	Bonacich P, Lloyd P 2001 Soc. Networks 23 191 Google Scholar
[20]	Brin S, Page L 1998 Comput. Netw. ISDN Syst. 30 107 Google Scholar
[21]	Martin T, Zhang X, Newman M E J 2014 Phys. Rev. E 90 052808 Google Scholar
[22]	Lü L, Chen D, Ren X L, et al. 2016 Phys. Rep. 650 1 Google Scholar
[23]	汪亭亭, 梁宗文, 张若曦 2023 72 048901 Google Scholar Wang T T, Liang Z W, Zhang R X 2023 Acta Phys. Sin. 72 048901 Google Scholar
[24]	Maji G, Namtirtha A, Dutta A, Malta M C 2020 Exp. Syst. Appl. 144 113092 Google Scholar
[25]	Liu J Q, Li X R, Dong J C 2021 Sci. China Technol. Sci. 64 451 Google Scholar
[26]	Liu Y, Zeng Q, Pan L, Tang M 2023 IEEE Trans. Netw. Sci. Eng. 10 2201 Google Scholar
[27]	Fan T, Lü L, Shi D, Zhou T 2021 Commun. Phys. 4 272 Google Scholar
[28]	阮逸润, 老松杨, 汤俊, 白亮, 郭延明 2022 71 176401 Google Scholar Ruan Y R, Lao S Y, Tang J, Bai L, Guo Y M 2022 Acta Phys. Sin. 71 176401 Google Scholar
[29]	Lung R I, Gaskó N, Suciu M A 2018 Scientometrics 117 1361 Google Scholar
[30]	Iacopini I, Petri G, Barrat A, Latora V 2019 Nat. Commun. 10 2485 Google Scholar
[31]	Battiston F, Cencetti G, Iacopini I, Latora V, Lucas M, Patania A, Young J G, Petri G 2020 Phys. Rep. 874 1 Google Scholar
[32]	de Arruda G F, Petri G, Moreno Y 2020 Phys. Rev. Res. 2 023032 Google Scholar
[33]	Wang W, Liu Q H, Liang J, Hu Y, Zhou T 2019 Phys. Rep. 820 1 Google Scholar
[34]	Li W Y, Xue X, Pan L, Lin T, Wang W 2022 Appl. Math. Comput. 412 126595
[35]	Fan J, Yin Q, Xia C, Perc M 2022 Proc. R. Soc. A. 478 20220059 Google Scholar
[36]	Estrada E, Ross G J 2018 J. Theor. Biol. 438 46 Google Scholar
[37]	Tudisco F, Higham D J 2021 Commun. Phys. 4 201 Google Scholar
[38]	Kovalenko K, Romance M, Vasilyeva E, et al. 2022 Chaos Solitons Fractals 162 112397 Google Scholar
[39]	Liu J G, Lin J H, Guo Q, Zhou T 2016 Sci. Rep. 6 21380 Google Scholar
[40]	Zeng Q, Liu Y, Tang M, Gong J 2021 Knowledge-Based Syst. 229 107365 Google Scholar
[41]	Li W, Nie Y, Li W, Chen X, Su S, Wang W 2022 Chaos 32 093135 Google Scholar
[42]	Wang H, Ma C, Chen H S, Lai Y C, Zhang H F 2022 Nat. Commun. 13 3043 Google Scholar
[43]	Génois M, Barrat A 2018 Epj Data Sci. 7 11 Google Scholar
[44]	Isella L, Stehlé J, Barrat A, Cattuto C, Pinton J F, Van den Broeck W 2011 J. Theor. Biol. 271 166 Google Scholar
[45]	Vanhems P, Barrat A, Cattuto C, Pinton J F, Khanafer N, Régis C, Kim B, Comte B, Voirin N 2013 PloS One 8 e73970 Google Scholar
[46]	Mastrandrea R, Fournet J, Barrat A 2015 PloS One 10 e0136497 Google Scholar

Cited By

图 1 单纯复形上SIR传播过程　(a)—(j) 传播过程; (k) 恢复过程

Figure 1. SIR spreading process on simplicial complex: (a)–(j) Spreading process; (k) recovery process.

DownLoad: Full-Size Img PowerPoint

图 2 节点传播中心性与传播影响力的散点图. $ \overline{{{\mathrm{SC}}}} $与$ \overline{{{\mathrm{S}}}} $分别表示归一化后的节点传播中心性与传播影响力　(a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13

Figure 2. Scatter plots of the spreaing centrality and spreaing influence of nodes. $ \overline{{{\mathrm{SC}}}} $ and $ {\overline{{\mathrm{S}}}} $ represent the normalized spreading centrality and spreading influence of nodes: (a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13.

DownLoad: Full-Size Img PowerPoint

图 3 各中心性的不准确函数　(a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13

Figure 3. Imprecisions of the centralities: (a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13

DownLoad: Full-Size Img PowerPoint

图 4 不同1阶单纯形传播速率$ \beta_1=\alpha\beta_1^{\mathrm{c}} $下, 节点各中心性与传播影响力的肯德尔相关系数　(a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13

Figure 4. Kendall’s tau correlation of the centralities and the spreading influence of nodes under different 1-simplex spreading rates $ \beta_1=\alpha\beta_1^{\mathrm{c}} $: (a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13

DownLoad: Full-Size Img PowerPoint

图 5 不同2阶单纯形传播速率$ \beta_2 $下, 节点各中心性与传播影响力的肯德尔相关系数　(a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13

Figure 5. Kendall’s tau correlation of the centralities and the spreading influence of nodes under different 2-simplex spreading rates $ \beta_2 $: (a) RSC; (b) SFSC; (c) InVS15; (d) LH10; (e) SFHH; (f) Thiers13

DownLoad: Full-Size Img PowerPoint

表 1 合成与真实单纯复形属性

Table 1. Properties of the synthetic and real simplicial complexes

网络	N	$\langle k_1\rangle$	$\langle k_2\rangle$	$\beta_1^{\mathrm{c}}$
RSC	2000	20	6	0.045
SFSC	5000	16	5	0.049
InVS15	213	20.19	7.94	0.040
LH10	72	15.94	13.04	0.042
SFHH	403	23.73	8.87	0.026
Thiers13	326	18.10	12.15	0.048

DownLoad: CSV

表 2 各中心性的Top-K准确率

Table 2. Top-K accuracy of centralities

网络	$K = 10$					$K = 20$					$K = 30$
网络	$ \mathrm{CI} $	$ \mathrm{NB} $	$ \mathrm{Deg} $	$ \mathrm{EVH} $	$ \mathrm{\mathrm{\mathrm{\mathrm{SC}\mathrm{ }}}} $	$ \mathrm{\mathrm{CI}} $	$ \mathrm{NB} $	$ \mathrm{Deg} $	$ \mathrm{\mathrm{EVH}} $	$ \mathrm{SC} $	$ \mathrm{CI} $	$ \mathrm{NB} $	$ \mathrm{\mathrm{Deg}} $	$ \mathrm{EVH} $	$ \mathrm{SC} $
RSC	0.60	0.60	0.50	0.60	0.60	0.70	0.75	0.60	0.75	0.75	0.70	0.77	0.57	0.77	0.73
SFSC	0.90	0.90	1.00	0.90	0.90	0.95	1.00	0.90	0.95	0.95	0.93	0.97	0.87	0.93	0.97
InVS15	0.40	0.90	0.50	0.70	0.90	0.65	0.90	0.70	0.90	0.90	0.77	0.97	0.70	0.83	0.90
LH10	0.10	1.00	0.90	0.90	0.90	0.55	0.95	0.90	0.95	0.95	0.87	1.00	0.93	0.97	1.00
SFHH	0.80	0.90	0.60	0.70	0.90	0.85	0.90	0.70	0.90	0.90	0.87	0.93	0.77	0.87	0.90
Thiers13	0.30	0.30	0.50	0.10	0.70	0.40	0.40	0.40	0.40	0.70	0.37	0.50	0.43	0.50	0.83

DownLoad: CSV

map

[1]	Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200 Google Scholar
[2]	Moreno Y, Nekovee M, Pacheco A F 2004 Phys. Rev. E 69 066130 Google Scholar
[3]	Motter A E 2004 Phys. Rev. Lett. 93 098701 Google Scholar
[4]	Li D, Fu B, Wang Y, Lu G, Berezin Y, Stanley H E, Havlin S 2014 Proc. Natl. Acad. Sci. 112 669
[5]	Kephart J O, Sorkin G B, Chess D M, White S R 1997 Sci. Am. 277 88
[6]	Hale T, Angrist N, Goldszmidt R, et al. 2023 Nat. Hum. Behav. 5 529 Google Scholar
[7]	Rocha Y M, de Moura G A, Desidério G A, et al. 2023 J. Public Health 31 1007 Google Scholar
[8]	Schäfer B, Witthaut D, Timme M, Latora V 2018 Nat. Commun. 9 1975 Google Scholar
[9]	任晓龙, 吕琳媛 2014 科学通报 59 1175 Google Scholar Ren X L, Lü L Y 2014 Sci. Bull. 59 1175 Google Scholar
[10]	Yang K C, Pierri F, Hui P M, Axelrod D, Torres-Lugo C, Bryden J, Menczer F 2021 Big Data Soc. 8 1
[11]	Nielsen B F, Simonsen L, Sneppen K 2021 Phys. Rev. Lett. 126 118301 Google Scholar
[12]	Freeman L C 1978 Soc. Networks 1 215 Google Scholar
[13]	Lü L, Zhou T, Zhang Q M, Stanley H E 2016 Nat. Commun. 7 10168 Google Scholar
[14]	Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nat. Phys. 6 888 Google Scholar
[15]	Morone F, Makse H A 2015 Nature 524 65 Google Scholar
[16]	Sabidussi G 1966 Psychometrika 31 581 Google Scholar
[17]	Freeman L C 1977 Sociometry 40 35 Google Scholar
[18]	Estrada E, Rodríguez-Velázquez J A 2005 Phys. Rev. E 71 056103 Google Scholar
[19]	Bonacich P, Lloyd P 2001 Soc. Networks 23 191 Google Scholar
[20]	Brin S, Page L 1998 Comput. Netw. ISDN Syst. 30 107 Google Scholar
[21]	Martin T, Zhang X, Newman M E J 2014 Phys. Rev. E 90 052808 Google Scholar
[22]	Lü L, Chen D, Ren X L, et al. 2016 Phys. Rep. 650 1 Google Scholar
[23]	汪亭亭, 梁宗文, 张若曦 2023 72 048901 Google Scholar Wang T T, Liang Z W, Zhang R X 2023 Acta Phys. Sin. 72 048901 Google Scholar
[24]	Maji G, Namtirtha A, Dutta A, Malta M C 2020 Exp. Syst. Appl. 144 113092 Google Scholar
[25]	Liu J Q, Li X R, Dong J C 2021 Sci. China Technol. Sci. 64 451 Google Scholar
[26]	Liu Y, Zeng Q, Pan L, Tang M 2023 IEEE Trans. Netw. Sci. Eng. 10 2201 Google Scholar
[27]	Fan T, Lü L, Shi D, Zhou T 2021 Commun. Phys. 4 272 Google Scholar
[28]	阮逸润, 老松杨, 汤俊, 白亮, 郭延明 2022 71 176401 Google Scholar Ruan Y R, Lao S Y, Tang J, Bai L, Guo Y M 2022 Acta Phys. Sin. 71 176401 Google Scholar
[29]	Lung R I, Gaskó N, Suciu M A 2018 Scientometrics 117 1361 Google Scholar
[30]	Iacopini I, Petri G, Barrat A, Latora V 2019 Nat. Commun. 10 2485 Google Scholar
[31]	Battiston F, Cencetti G, Iacopini I, Latora V, Lucas M, Patania A, Young J G, Petri G 2020 Phys. Rep. 874 1 Google Scholar
[32]	de Arruda G F, Petri G, Moreno Y 2020 Phys. Rev. Res. 2 023032 Google Scholar
[33]	Wang W, Liu Q H, Liang J, Hu Y, Zhou T 2019 Phys. Rep. 820 1 Google Scholar
[34]	Li W Y, Xue X, Pan L, Lin T, Wang W 2022 Appl. Math. Comput. 412 126595
[35]	Fan J, Yin Q, Xia C, Perc M 2022 Proc. R. Soc. A. 478 20220059 Google Scholar
[36]	Estrada E, Ross G J 2018 J. Theor. Biol. 438 46 Google Scholar
[37]	Tudisco F, Higham D J 2021 Commun. Phys. 4 201 Google Scholar
[38]	Kovalenko K, Romance M, Vasilyeva E, et al. 2022 Chaos Solitons Fractals 162 112397 Google Scholar
[39]	Liu J G, Lin J H, Guo Q, Zhou T 2016 Sci. Rep. 6 21380 Google Scholar
[40]	Zeng Q, Liu Y, Tang M, Gong J 2021 Knowledge-Based Syst. 229 107365 Google Scholar
[41]	Li W, Nie Y, Li W, Chen X, Su S, Wang W 2022 Chaos 32 093135 Google Scholar
[42]	Wang H, Ma C, Chen H S, Lai Y C, Zhang H F 2022 Nat. Commun. 13 3043 Google Scholar
[43]	Génois M, Barrat A 2018 Epj Data Sci. 7 11 Google Scholar
[44]	Isella L, Stehlé J, Barrat A, Cattuto C, Pinton J F, Van den Broeck W 2011 J. Theor. Biol. 271 166 Google Scholar
[45]	Vanhems P, Barrat A, Cattuto C, Pinton J F, Khanafer N, Régis C, Kim B, Comte B, Voirin N 2013 PloS One 8 e73970 Google Scholar
[46]	Mastrandrea R, Fournet J, Barrat A 2015 PloS One 10 e0136497 Google Scholar

[1]	Cui Jun-Ying, Xu Shu-Qi, Na Xu, Pan Li-Ming, Lü Lin-Yuan. Supply chain research based on complex network theory. Acta Physica Sinica, 2024, 73(19): 198901. doi: 10.7498/aps.73.20240702
[2]	Liu Bo, Zeng Yu-Jie, Yang Rong-Mei, Lü Lin-Yuan. Fundamental statistics of higher-order networks: a survey. Acta Physica Sinica, 2024, 73(12): 128901. doi: 10.7498/aps.73.20240270
[3]	Luo Kai-Ming, Guan Shu-Guang, Zou Yong. Reconstruction of simplex structures based on phase synchronization dynamics. Acta Physica Sinica, 2024, 73(12): 120501. doi: 10.7498/aps.73.20240334
[4]	Ruan Yi-Run, Lao Song-Yang, Tang Jun, Bai Liang, Guo Yan-Ming. Node importance ranking method in complex network based on gravity method. Acta Physica Sinica, 2022, 71(17): 176401. doi: 10.7498/aps.71.20220565
[5]	Tan Suo-Yi, Qi Ming-Ze, Wu Jun, Lu Xin. Link predictability of complex network from spectrum perspective. Acta Physica Sinica, 2020, 69(8): 088901. doi: 10.7498/aps.69.20191817
[6]	Kong Jiang-Tao, Huang Jian, Gong Jian-Xing, Li Er-Yu. Evaluation methods of node importance in undirected weighted networks based on complex network dynamics models. Acta Physica Sinica, 2018, 67(9): 098901. doi: 10.7498/aps.67.20172295
[7]	Su Zhen, Gao Chao, Li Xiang-Hua. Analysis of the effect of node centrality on diffusion mode in complex networks. Acta Physica Sinica, 2017, 66(12): 120201. doi: 10.7498/aps.66.120201
[8]	Ruan Yi-Run, Lao Song-Yang, Wang Jun-De, Bai Liang, Hou Lü-Lin. An improved evaluating method of node spreading influence in complex network based on information spreading probability. Acta Physica Sinica, 2017, 66(20): 208901. doi: 10.7498/aps.66.208901
[9]	Han Zhong-Ming, Chen Yan, Li Meng-Qi, Liu Wen, Yang Wei-Jie. An efficient node influence metric based on triangle in complex networks. Acta Physica Sinica, 2016, 65(16): 168901. doi: 10.7498/aps.65.168901
[10]	Hu Qing-Cheng, Zhang Yong, Xu Xin-Hui, Xing Chun-Xiao, Chen Chi, Chen Xin-Hua. A new approach for influence maximization in complex networks. Acta Physica Sinica, 2015, 64(19): 190101. doi: 10.7498/aps.64.190101
[11]	Han Zhong-Ming, Wu Yang, Tan Xu-Sheng, Duan Da-Gao, Yang Wei-Jie. Ranking key nodes in complex networks by considering structural holes. Acta Physica Sinica, 2015, 64(5): 058902. doi: 10.7498/aps.64.058902
[12]	Ren Zhuo-Ming, Liu Jian-Guo, Shao Feng, Hu Zhao-Long, Guo Qiang. Analysis of the spreading influence of the nodes with minimum K-shell value in complex networks. Acta Physica Sinica, 2013, 62(10): 108902. doi: 10.7498/aps.62.108902
[13]	Yu Hui, Liu Zun, Li Yong-Jun. Key nodes in complex networks identified by multi-attribute decision-making method. Acta Physica Sinica, 2013, 62(2): 020204. doi: 10.7498/aps.62.020204
[14]	Liu Jian-Guo, Ren Zhuo-Ming, Guo Qiang, Wang Bing-Hong. Node importance ranking of complex networks. Acta Physica Sinica, 2013, 62(17): 178901. doi: 10.7498/aps.62.178901
[15]	Liu Jin-Liang. Research on synchronization of complex networks with random nodes. Acta Physica Sinica, 2013, 62(4): 040503. doi: 10.7498/aps.62.040503
[16]	Zhou Xuan, Zhang Feng-Ming, Zhou Wei-Ping, Zou Wei, Yang Fan. Evaluating complex network functional robustness by node efficiency. Acta Physica Sinica, 2012, 61(19): 190201. doi: 10.7498/aps.61.190201
[17]	LÜ Ling, Liu Shuang, Zhang Xin, Zhu Jia-Bo, Shen Na, Shang Jin-Yu. Spatiotemporal chaos anti-synchronization of a complex network with different nodes. Acta Physica Sinica, 2012, 61(9): 090504. doi: 10.7498/aps.61.090504
[18]	Zhou Xuan, Zhang Feng-Ming, Li Ke-Wu, Hui Xiao-Bin, Wu Hu-Sheng. Finding vital node by node importance evaluation matrix in complex networks. Acta Physica Sinica, 2012, 61(5): 050201. doi: 10.7498/aps.61.050201
[19]	Lü Ling, Zhang Chao. Chaos synchronization of a complex network with different nodes. Acta Physica Sinica, 2009, 58(3): 1462-1466. doi: 10.7498/aps.58.1462
[20]	Li Ji, Wang Bing-Hong, Jiang Pin-Qun, Zhou Tao, Wang Wen-Xu. Growing complex network model with acceleratingly increasing number of nodes. Acta Physica Sinica, 2006, 55(8): 4051-4057. doi: 10.7498/aps.55.4051

Catalog

Vol. 73, Issue 4 - 2024-02-20

Metrics

Abstract views: 2906
PDF Downloads: 138
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板