Rumor propagation dynamics in social networks under the influence of time delay and diffusion

Wang Nan; Xiao Min; Jiang Hai-Jun; Huang Xia

doi:10.7498/aps.71.20220726

Abstract

Rumors in social networks are often referred to as infectious diseases of the Internet, because rumors spreading in networks feature strong concealment, fast transmission speed and wide spread. With the development of mobile devices, online rumors nowadays are far more harmful than before. Rumors in social networks show completely different spatiotemporal dynamics from traditional rumor spreading dynamics. A social network rumor spreading model with considering both reaction diffusion and fermentation time delay is proposed in this paper. The effects of spatial diffusion and time delay on rumor spreading in online social networks are studied. Firstly, the existence of equilibrium point of the reaction-diffusion rumor spreading model is analyzed, and the basic regeneration number $R_{0}$ is calculated. When $R_{0} < 1$, the rumor stops spreading and disappears in social networks; when $R_{0}>1$, the rumor persists in social networks. Secondly, the local stability of the rumor spreading equilibrium is investigated by using the Roth-Hurwitz stability criterion, and the influence of diffusion on the system stability is discussed. When the diffusion is introduced into a stable rumor spreading model without time delay, the model becomes unstable, indicating that the Turing instability is caused by diffusion. Thirdly, the Hopf bifurcation condition of the rumor spreading model is established by selecting the time delay τ as the bifurcation parameter, and the expression of bifurcation threshold $\tau_{0}$ is given. When $\tau < \tau_{0}$, the rumor propagation model with diffusion term is stable; when $\tau>\tau_{0}$, the model loses the stability and the Hopf bifurcation occurs. The numerical simulation results show that both diffusion and time delay play an important role in the dynamic evolution of rumor spreading. At the same time, the influence of the crowding degree of spreaders on rumor propagation is also simulated. As the crowding gets worse and worse, the rumor refuting effect weakens, the bifurcation threshold $\tau_{0}$ decreases, and the propagation peak increases. Therefore, it is important to build an excellent social network environment to supervise the rumors that are still in the fermentation stage, improve the timeliness of the release of rumor refuting information, and strengthen the refuting of rumors among key groups. This paper breaks through the limitation considering only the time evolution, explores the spatiotemporal spreading law of rumor in real society, and provides a new perspective and idea for governing the rumor spreading.

Keywords:

Authors and contacts

1.
College of Automation & College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
2.
College of Mathematics and Systems Science, Xinjiang University, Urumchi 830047, China
3.
College of Electrical and Automation Engineering, Shandong University of Science and Technology, Qingdao 266590, China

Corresponding author: Xiao Min, candymanxm2003@aliyun.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 62073172, U1703262, 61973199)

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References

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图 1 当$\tau = 0, \;d_{1}=d_{2}=d_{3} = 0$ 时, 模型(2)在$ E^{\ast} $处是局部渐近稳定的

Figure 1. Model (2) is locally asymptotically stable at $ E^{\ast} $ when $ \tau = 0, d_{1}=d_{2}=d_{3} = 0 $

DownLoad: Full-Size Img PowerPoint

图 2 当$\tau = 0,\; d_{1} = 6,\; d_{2} = 0.01, \;d_{3} = 0.01$ 时, 模型(2)在$ E^{\ast} $处是不稳定的

Figure 2. Model (2) is unstable at $ E^{\ast} $ when $\tau = 0,\; d_{1} = 6,\; d_{2} = 0.01, \;d_{3} = 0.01$

DownLoad: Full-Size Img PowerPoint

图 3 Turing不稳定发生时模型(2)的Turing斑图

Figure 3. Turing patterns of model (2) when Turing instability occurs

DownLoad: Full-Size Img PowerPoint

图 4 当$ \tau = 2.6 < \tau_{0} = 2.6332 $时模型(2) 的波形图

Figure 4. Waveform plots of model (2) with $ \tau = 2.6 < \tau_{0} = 2.6332 $

DownLoad: Full-Size Img PowerPoint

图 5 当$ \tau = 2.655 > \tau_{0} = 2.6332 $时模型(2)的波形图

Figure 5. Waveform plots of model (2) with $ \tau = 2.655 > \tau_{0} = 2.6332 $

DownLoad: Full-Size Img PowerPoint

图 6 模型(2)在位置x = 4处的相图 (a) $ \tau = 2.6 $; (b) $ \tau = 2.655 $

Figure 6. Phase diagram of model (2) at x = 4: (a) $ \tau = 2.6 $; (b) $ \tau = 2.655 $

DownLoad: Full-Size Img PowerPoint

图 7 不同α取值时I的波形图

Figure 7. Waveform of I with different values of α

DownLoad: Full-Size Img PowerPoint

表 1 不同α对应的平衡点与分岔阈值

Table 1. Equilibrium point and bifurcation threshold for different α

饱和因子α	平衡点$(S^{\ast}, I^{\ast}, R^{\ast})$	分岔阈值$\tau_{0}$
0.1	(1.1615, 0.6479, 0.8572)	2.8331
0.3	(1.1074, 0.7040, 0.8552)	2.7589
0.5	(1.0596, 0.7584, 0.8487)	2.7048
0.7	(1.0182, 0.8094, 0.8390)	2.6436
0.9	(0.9833, 0.8560, 0.8274)	2.5143

DownLoad: CSV

map

[1]	Liu X, Li T, Xu H, Liu W 2019 Physica A 514 497 Google Scholar
[2]	Hui H, Zhou C, Lu X, Li J 2020 Nonlinear Dyn. 101 11 Google Scholar
[3]	王祁月, 刘润然, 贾春晓 2021 70 068902 Google Scholar Wang Q Y, Liu R R, Jia C X 2021 Acta Phys. Sin. 70 068902 Google Scholar
[4]	Zhou D, Han W, Wang Y 2015 J. Comput. Res. Dev. 52 156 Google Scholar
[5]	Youssef B, Gregory Z, Judicael R, Bruno V 2019 Simulation 95 411 Google Scholar
[6]	李瑾颉, 吴联仁, 齐佳音, 闫强 2017 电子与信息学报 39 785 Google Scholar Li J J, Wu L R, Qi J Y, Yan Q 2017 J. Electron. Inf. Technol. 39 785 Google Scholar
[7]	Junior V V, Rodriguez P M, Speroto A 2021 J. Stat. Mech.: Theory Exp. 12 123403 Google Scholar
[8]	张菊平, 郭昊明, 荆文君, 靳祯 2019 68 150501 Google Scholar Zhang J P, Guo H M, Jing W J, Jin Z 2019 Acta Phys. Sin. 68 150501 Google Scholar
[9]	Zhu L, Wang B 2020 Inf. Sci. 526 1 Google Scholar
[10]	Jia P, Wang C, Zhang G, Ma J 2019 Physica A 524 342 Google Scholar
[11]	Huo L, Chen S, Zhao L 2021 Physica A 571 125828 Google Scholar
[12]	Zanette D H 2001 Phys. Rev. E 64 050901 Google Scholar
[13]	Moreno Y, Pacheco A F, Nekovee M 2004 Phys. Rev. E 69 066130 Google Scholar
[14]	Zhou J, Liu Z H, Li B M 2007 Phys. Lett. A 368 458 Google Scholar
[15]	张芳, 司光亚, 罗批 2009 复杂系统与复杂性科学 6 1 Google Scholar Zhang F, Si G Y, Luo P 2009 Complex Systems and Complexity Science 6 1 Google Scholar
[16]	Zhao L J, Wang Q, Cheng J J, Chen Y C, Wang J J, Huang W 2011 Physica A 390 2619 Google Scholar
[17]	顾亦然, 夏玲玲 2012 61 238701 Google Scholar Gu Y R, Xia L L 2012 Acta Phys. Sin. 61 238701 Google Scholar
[18]	万佑红, 王小初 2016 计算机应用 36 2381 Google Scholar Wan Y H, Wang X C 2016 Journal of Computer Applications 36 2381 Google Scholar
[19]	朱霖河, 李玲 2020 69 020501 Google Scholar Zhu L H, Li L 2020 Acta Phys. Sin. 69 020501 Google Scholar
[20]	Ruan Z Y, Yu B, Shu X C, Zhang Q P, Xuan Q 2020 Chaos 30 083101 Google Scholar
[21]	Chen X L, Wang N 2020 Sci. Rep. 10 1 Google Scholar
[22]	Ahmed N, Korkamaz A, Rehman M A, Rafiq M, Ali M, Ahmad M O 2020 Int. J. Comput. Math. 98 517 Google Scholar
[23]	Hattaf K, Yousfi N 2016 Comput. Math. Appl. 72 2741 Google Scholar
[24]	赵洪涌, 朱霖河 2015 南京航空航天大学学报 47 332 Google Scholar Zhao H Y, Zhu L H 2015 Journal of Nanjing University of Aeronautics & Astronautics 47 332 Google Scholar
[25]	Zhu L, Huang X, Liu Y, Zhang Z 2021 J. Math. Anal. Appl. 493 124539 Google Scholar
[26]	Tan W, Yu W W, Hayat T, Alsaadi F, Fardoun H M 2018 Int. J. Bifurcation and Chaos 28 1830029 Google Scholar
[27]	Peter F P, John F, Jakob H 2019 Phys. Rev. E 99 022302 Google Scholar
[28]	Van D D, Watmough J 2002 Math. Biosci. 180 29 Google Scholar

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