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In this paper, the influence of opinion dynamics on rumor propagation is studied by introducing the mechanism of stifler’s comments on rumors into the susceptible-infected-recovered (SIR) model. In this model, individuals can form a group together with their direct neighbors, through which the spreader can spread rumors, and the stiflers can express opinions and refute the rumors. The mechanism of rumor refuting can not only reduce the acceptance rate of the ignorant to rumor, but also increase the transition probability of the spreader to be a stifler. In this paper, we use the Erdös-Rényi (ER) random network, scale-free network and real social network as the underlying interaction structure to study the influence of stifler’s silence probability on the rumor spreading dynamics. First of all, we find that the process of rumor propagation can be roughly divided into two stages, i.e., the early stage of free propagation of rumors and the later stage of checks and balances between the stiflers and the spreaders, respectively. Secondly, it is found that the rumor will break out with the increase of stifler’s silence probability. Under a threshold of rumor outbreak, the increase of silence probability will not lead the number of spreaders to significantly increase, but will cause more ignorance to perceive the rumor and quickly turn into spreaders, and then change into stiflers under the guidance of other stiflers. When the silence probability reaches a threshold, the stiflers will not be able to control the spread of rumors, which will lead the number of stiflers to decrease and the rumors to break out. Finally, the early stage of rumor propagation in scale-free networks is shorter than that of random network, which makes rumor more difficult to break out. Our model comprehensively considers the influence of opinion dynamics on the spreading of rumors and more realistically simulates the rumor diffusion process, which provides a useful insight for the rumor control in real-world social networks.
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Keywords:
- rumor propagation /
- online social network /
- opinion dynamics /
- dispel rumors
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Google Scholar
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[13] Kermack W O, Mckendrick A G 1991 Bull. Mathemat. Biol. 53 5787
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Google Scholar
[15] Moreno Y, Nekovee M, Vespignani A 2004 Phys. Rev. E 69 055101
Google Scholar
[16] Borge-Holthoefer J, Moreno Y 2012 Phys. Rev. E 85 026116
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[27] Xia L L, Jiang G P, Song B, Song Y R 2015 Physica A 437 295
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Shan X G, Zhu Y, Sun M 2015 Rep. on Stud. China Med. Devel. 2015 168
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Google Scholar
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图 1 谣言传播示意图 (a) 网络有三种状态节点, S状态、I状态和R状态, 图中蓝色圆圈表示其中1个传播者I与其邻居所构成的群; (b), (c)传播者向在以自己为中心的群中传播信息; 在受到抑制者的影响的情况下, S状态节点以一定概率转换成I状态节点, I状态节点以一定概率转换成R状态节点; (d) 系统到达稳态, 传播停止
Figure 1. Schematic diagram of rumor propagation. (a) The network is composed of N nodes with three states S, I and R. The blue circle in the figure represents the group formed by one of the spreaders I and its neighbors. (b), (c) Spreaders spread the rumor to their self-centered groups. Influenced by the stiflers, S state node turns into I state node with a certain probability, and I state node turns into R state node with a certain probability. (d) The system reaches to a steady state and the spreading process ends.
图 2 (a) ER网络上SIR三种状态的节点比例随时间的演化图, 沉默概率
$\gamma = {{0}}{{.4}}$ ; (b) ER网络上稳态SIR节点比例随$\gamma $ 值的变化图, 图中曲线来自于网络规模为N = 1 × 106, 平均度为10 的ER随机网络, 初始传播者和抑制者的比例均设为0.1%. 图(b) 中的数据来自于演化时间t = 4 × 106时的结果Figure 2. (a) Time evolution of the fractions of nodes with different states on ER network, where the silence probability
$\gamma $ is equal to 0.4; (b) the steady fractions of nodes with different states as a function of$\gamma $ on ER networks. The curves in the figure come from the ER random network with a size N = 1 × 106 and an average degree of 10. The fractions of both the initial spreaders and the stiflers are set as 0.1%. The data in panel (b) are from the simulation results for t = 4 × 106.
图 3 ER网络上不同沉默概率
$\gamma $ = 0.3, 0.6, 0.9下SIR三种不同状态节点的比例随时间演化图 (a) S态; (b) I态; (c) R态; 初始传播者和抑制者的比例均为0.1%, ER网络的平均度为10Figure 3. Time evolution of the fractions of nodes with different states on ER random networks with different silencing probability
$\gamma $ = 0.3, 0.6 and 0.9: (a) State S; (b) state I; (c) state R. The initial fractions of spreaders and the stiflers are both 0.1%, and the average degree of the ER network is 10.
图 4 (a)无标度网络上SIR三种状态的节点比例随时间的演化图, 沉默概率
$\gamma = {{0}}{{.4}}$ ; (b)无标度网络上稳态SIR节点比例随$\gamma $ 值的变化图, 图中曲线来自于网络规模为N = 1 × 106, 平均度为10的无标度网络, 初始传播者和抑制者的比例均为0.1%, 图(b) 中的数据来自于演化时间t = 4 × 106时的结果Figure 4. (a) Time evolution of the fractions of nodes with different states on scale-free networks, where the silence probability
$\gamma $ is equal to 0.4; (b) the steady fractions of nodes with different states as a function of$\gamma $ on scale-free network. The curves in the figure come from the scale-free network with a network size of N = 1 × 106 and an average degree of 10. The fractions of both the initial spreaders and the stiflers are set as 0.1%. The data in panel (b) are from the simulation results for t = 4 × 106.
图 5 无标度网络上不同沉默概率
$\gamma $ = 0.3, 0.6, 0.9下SIR三种状态节点的比例随时间演化图 (a) S态; (b) I态; (c) R态; 初始传播者和抑制者的比例均为0.1%, 网络平均度为10Figure 5. Time evolution of the fractions of nodes with different states on scale-free networks with different silencing probability
$\gamma $ = 0.3, 0.6 and 0.9: (a) State S; (b) state I; (c) state R. The initial fractions of spreaders and the stiflers are both 0.1%, and the average degree of the scale-free network is 10.
图 6 Facebook网络(a)和Gowalla网络(b)在双对数坐标下的累积度分布P(k)
Figure 6. Cumulative degree distribution of Facebook network (a) and Gowalla network (b) in logarithmic coordinates.
图 7 Facebook网络(a)和Gowalla网络(b)稳态SIR三种节点比例随不同
$\gamma $ 值的变化; 在数值模拟中, 初始传播者和抑制者的比例均设为0.1%; 图(b) 中的数据来自于演化时间t = 4 × 106时的结果Figure 7. Steady fraction of nodes with different states as functions of
$\gamma $ on Facebook network (a) and Gowalla network (b). In the numerical simulations, the initial fractions of spreaders and the stiflers are both 0.1%. The data in panel (b) are from the simulation results for t = 4 × 106. -
[1] Skvoretz J, Faust K, Fararo T J 1996 J. Math. Sociol. 21 57
Google Scholar
[2] Weng L L, Menczer F, Ahn Y Y 2013 Sci. Rep. 3 2522
Google Scholar
[3] Lazer D M J, Baum M A, Benkler Y, et al. 2018 Science 359 6380
Google Scholar
[4] Castellano C, Fortunato S, Loreto V 2009 Rev. Mod. Phys. 81 591
Google Scholar
[5] Moreno Y, Nekovee M, Pachec A F 2004 Phys. Rev. E 69 066130
Google Scholar
[6] Zanette D H 2002 Phys. Rev. E 65 041908
Google Scholar
[7] Yang J, McAuley J, Leskovec J 2013 IEEE 13th International Conference on Data Mining Texas, USA, December 7–10, 2013 p1151
[8] Moreno Y, Gómez J B, Pacheco A F 2003 Phys. Rev. E 68 035103
Google Scholar
[9] Shu P P, Gao L, Zhao P C, Wang W, Stanley H E 2017 Sci. Rep. 7 44669
Google Scholar
[10] Guilbeault D, Becker J, Centola D 2018 Complex Spreading Phenomena in Social Systems (Berlin: Springer) pp3−25
[11] Daley D J, Kendall D J 1964 Nature 204 1118
Google Scholar
[12] Maki D P, Thompson M 1973 Mathematical Models and Applications: With Emphasis on the Social, Life, and Management Sciences (New Jersey: Prentice-Hall) p492
[13] Kermack W O, Mckendrick A G 1991 Bull. Mathemat. Biol. 53 5787
[14] Liu Z H, Lai Y C, Ye N 2003 Phys. Rev. E 67 031911
Google Scholar
[15] Moreno Y, Nekovee M, Vespignani A 2004 Phys. Rev. E 69 055101
Google Scholar
[16] Borge-Holthoefer J, Moreno Y 2012 Phys. Rev. E 85 026116
Google Scholar
[17] Chen Y, Wang W, Feng J P, Lu Y, Gong X Q 2020 PLoS One 15 e0229201
Google Scholar
[18] Shah D, Zaman T 2011 IEEE Trans. Inf. Theory 57 8
Google Scholar
[19] Sun Y, Ma L, Zeng A, Wang W X 2016 Sci. Rep. 6 38865
Google Scholar
[20] Wu D Y, Tang M, Liu Z H, Lai Y C 2020 Commun. Nonlinear Sci. Numer. Simul. 90 105403
Google Scholar
[21] Khanjanianpak M, Azimi-Tafreshi N, Castellano C 2020 Phys. Rev. E 101 062306
Google Scholar
[22] Granell C, Gómez S, Arenas A 2014 Phys. Rev. E 90 012808
Google Scholar
[23] Grassberger P, Chen L, Ghanbarnejad F, Cai W R 2016 Phys. Rev. E 93 042316
Google Scholar
[24] Sahneh F D, Scoglio C M 2012 IEEE 51st IEEE Conference on Decision and Control Maui, USA, December 10–13, 2012 p1657
[25] Pan Y H, Yan Z J 2018 Chaos 28 063123
Google Scholar
[26] Zhao L J, Wang J J, Chen Y C, Wang Q, Cheng J J, Cui H X 2012 Physica A 391 2444
Google Scholar
[27] Xia L L, Jiang G P, Song B, Song Y R 2015 Physica A 437 295
Google Scholar
[28] Soriano-Paños D, Guo Q, Latora V, Gómez-Gardeñes J 2019 Phys. Rev. E 99 062311
[29] Sznajd-Weron K, Sznajd J 2000 Int. J. Mod. Phys. C 11 1157
Google Scholar
[30] Holley R A, Liggett T M 1975 Ann. Prob. 3 643
Google Scholar
[31] Deffuant G, Neau D, Amblard F, Weisbuch G 2001 Adv. Complex Syst. 3 11
[32] Hegselmann R, Krause U 2002 J. Arti. Soc. Social Simul. 5 3
[33] Meng X F, Van Gorder R A, Porter M A 2018 Phys. Rev. E 97 022312
[34] Ghose A, Ipeirotis P G 2010 IEEE Trans. Know. Data Eng. 23 1498
Google Scholar
[35] 张亮, 杨闪, 李霞 2017 系统工程 35 82
Google Scholar
Zhang L, Yang S, Li X 2017 Systems Engineering 35 82
Google Scholar
[36] Petra P, Ewa D S 2019 J. Soc. Political Psy. 7 2
Google Scholar
[37] Li R Q, Li Y W, Meng Z Y, Song Y R, Jiang G P 2020 IEEE Access 8 63065
Google Scholar
[38] 单学刚, 朱燕, 孙敏 2015 中国媒体发展研究报告 2015 168
Shan X G, Zhu Y, Sun M 2015 Rep. on Stud. China Med. Devel. 2015 168
[39] Sudbury A 1985 J. Appl. Prob. 22 443
Google Scholar
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