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September  2018, 13(3): 515-530. doi: 10.3934/nhm.2018023

SIR Rumor spreading model with trust rate distribution

Bum Il Hong 1, , Nahmwoo Hahm 1,2, and  Sun-Ho Choi 1,2,,

1. 

Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 446-701, Korea
2. 

Department of Mathematics, Incheon National University, Incheon, 406-772, Korea

* Corresponding author: Sun-Ho Choi

Received  December 2017 Revised  March 2018 Published  July 2018

Fund Project: The research of B. I. Hong and S.-H. Choi was supported by Korea Electric Power Corporation(Grant number:R18XA02).

In this paper, we study a rumor spreading model in which several types of ignorants exist with trust rate distributions $λ_i $, $≤ i≤ N$. We rigorously show the existence of a threshold on a momentum type initial quantity related to rumor outbreak occurrence regardless of the total initial population. We employ a steady state analysis to obtain the final size of the rumor. Using numerical simulations, we demonstrate the analytical result in which the threshold phenomenon exists for rumor size and discuss interaction between the ignorants of several types of trust rates.

Keywords: Rumor spreading model, SIR, rumor outbreak, stability.
Mathematics Subject Classification: Primary: 34D20, 34C12; Secondary: 34C23.
Citation: Bum Il Hong, Nahmwoo Hahm, Sun-Ho Choi. SIR Rumor spreading model with trust rate distribution. Networks and Heterogeneous Media, 2018, 13 (3) : 515-530. doi: 10.3934/nhm.2018023
References:
[1]

P. Bordia and N. DiFonzo, Problem solving in social interactions on the Internet: Rumor as social cognition, Social Psychology Quarterly, 67 (2004), 33-49.  doi: 10.1177/019027250406700105.

[2]

G. Chen, H. Shen, T. Ye, G. Chen and N. Kerr, A kinetic model for the spread of rumor in emergencies, Discrete Dynamics in Nature and Society, 2013 (2013), Art. ID 605854, 8 pp.

[3]

D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0.

[4]

D. J. Daley and D. G. Kendall, Stochastic rumours, IMA Journal of Applied Mathematics, 1 (1965), 42-55.  doi: 10.1093/imamat/1.1.42.

[5]

N. Fountoulakis, K. Panagiotou and T. Sauerwald, Ultra-fast rumor spreading in social networks, Proceedings of the Twenty-Third Annual ACM-SIAM symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2012, 1642–1660. doi: 10.1137/1.9781611973099.130.

[6]

J. Gu, W. Li and X. Cai, The effect of the forget-remember mechanism on spreading, European Physical Journal B, 62 (2008), 247-255.  doi: 10.1140/epjb/e2008-00139-4.

[7]

V. Isham, S. Harden and M. Nekovee, Stochastic epidemics and rumours on finite random networks, Physica A, 389 (2010), 561-576.  doi: 10.1016/j.physa.2009.10.001.

[8]

M. E. Jaeger, S. Anthony and R. L. Rosnow, Who hears what from whom and with what effect:A study of rumor, Personality and Social Psychology Bulletin, 6 (1980), 473-478. 

[9]

J. Jiang, S. Gong and B. He, Dynamical behavior of a rumor transmission model with Holling-type Ⅱ functional response in emergency event, Physica A, 450 (2016), 228-240.  doi: 10.1016/j.physa.2015.12.143.

[10]

A. J. Kimmel and A.-F. Audrain-Pontevia, Analysis of commercial rumors from the perspective of marketing managers: Rumor prevalence, effects, and control tactics, Journal of Marketing Communications, 16 (2010), 239-253.  doi: 10.1080/13527260902884433.

[11]

D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.

[12]

M. McDonald, O. Suleman, S. Williams, S. Howison and N. F. Johnson, Impact of unexpected events, shocking news, and rumors on foreign exchange market dynamics, Physical Review E, 77 (2008), 046110. doi: 10.1103/PhysRevE.77.046110.

[13]

Y. Moreno, M. Nekovee and A. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E, 69 (2004), 066130. doi: 10.1103/PhysRevE.69.066130.

[14]

M. Nagao, K. Suto and A. Ohuchi, A media information analysis for implementing effective countermeasure against harmful rumor, Journal of Physics, Conference Series, 221 (2010), 012004. doi: 10.1088/1742-6596/221/1/012004.

[15]

M. Nekovee, Y. Moreno, G. Bianconi and M. Marsili, Theory of rumour spreading in complex social networks, Physica A, 374 (2007), 457-470.  doi: 10.1016/j.physa.2006.07.017.

[16]

R. L. Rosnow, J. H. Yost and J. L. Esposito, Belief in rumor and likelihood of rumor transmission, Language & Communication, 6 (1986), 189-194.  doi: 10.1016/0271-5309(86)90022-4.

[17]

A. Sudbury, The proportion of population never hearing a rumour, Journal of Applied Probability, 22 (1985), 443-446.  doi: 10.2307/3213787.

[18]

S. A. Thomas, Lies, damn lies, and rumors: An analysis of collective efficacy, rumors, and fear in the wake of Katrina, Sociological Spectrum, 27 (2007), 679-703.  doi: 10.1080/02732170701534200.

[19]

Y.-Q. Wang, X.-Y. Yang, Y.-L. Han and X.-A. Wang, Rumor Spreading Model with Trust Mechanism in Complex Social Networks, Communications in Theoretical Physics, 59 (2013), 510-516. 

[20]

J. Wang, L. Zhao and R. Huang, 2SI2R rumor spreading model in homogeneous networks, Physica A, 413 (2014), 153-161.  doi: 10.1016/j.physa.2014.06.053.

[21]

D. J. Watts and S. H. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), 440-442. 

[22]

Y. Zan, J. Wua, P. Li and Q. Yua, SICR rumor spreading model in complex networks: Counterattack and self-resistance, Physica A, 405 (2014), 159-170.  doi: 10.1016/j.physa.2014.03.021.

[23]

D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901. doi: 10.1103/PhysRevE.64.050901.

[24]

D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908. doi: 10.1103/PhysRevE.65.041908.

[25]

L. Zhao, H. Cui, X. Qiu, X. Wang and J. Wang, SIR rumor spreading model in the new media age, Physica A, 392 (2013), 995-1003.  doi: 10.1016/j.physa.2012.09.030.

[26]

L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng and H. Cui, SIHR rumor spreading model in social networks, Physica A, 391 (2012), 2444-2453. 

[27]

L. Zhao, Q. Wang, J. Cheng, Y. Chen, J. Wang and W. Huang, Rumor spreading model with consideration of forgetting mechanism: A case of online blogging LiveJournal, Physica A, 390 (2011), 2619-2625.  doi: 10.1016/j.physa.2011.03.010.

[28]

L. Zhao, W. Xie, H. O. Gao, X. Qiu, X. Wang and S. Zhang, A rumor spreading model with variable forgetting rate, Physica A, 392 (2013), 6146-6154.  doi: 10.1016/j.physa.2013.07.080.

[29]

L. Zhu, H. Zhao and H. Wang, Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms, Information Sciences, 349/350 (2016), 119-136.  doi: 10.1016/j.ins.2016.02.031.

show all references

References:
[1]

P. Bordia and N. DiFonzo, Problem solving in social interactions on the Internet: Rumor as social cognition, Social Psychology Quarterly, 67 (2004), 33-49.  doi: 10.1177/019027250406700105.

[2]

G. Chen, H. Shen, T. Ye, G. Chen and N. Kerr, A kinetic model for the spread of rumor in emergencies, Discrete Dynamics in Nature and Society, 2013 (2013), Art. ID 605854, 8 pp.

[3]

D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature, 204 (1964), 1118. doi: 10.1038/2041118a0.

[4]

D. J. Daley and D. G. Kendall, Stochastic rumours, IMA Journal of Applied Mathematics, 1 (1965), 42-55.  doi: 10.1093/imamat/1.1.42.

[5]

N. Fountoulakis, K. Panagiotou and T. Sauerwald, Ultra-fast rumor spreading in social networks, Proceedings of the Twenty-Third Annual ACM-SIAM symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2012, 1642–1660. doi: 10.1137/1.9781611973099.130.

[6]

J. Gu, W. Li and X. Cai, The effect of the forget-remember mechanism on spreading, European Physical Journal B, 62 (2008), 247-255.  doi: 10.1140/epjb/e2008-00139-4.

[7]

V. Isham, S. Harden and M. Nekovee, Stochastic epidemics and rumours on finite random networks, Physica A, 389 (2010), 561-576.  doi: 10.1016/j.physa.2009.10.001.

[8]

M. E. Jaeger, S. Anthony and R. L. Rosnow, Who hears what from whom and with what effect:A study of rumor, Personality and Social Psychology Bulletin, 6 (1980), 473-478. 

[9]

J. Jiang, S. Gong and B. He, Dynamical behavior of a rumor transmission model with Holling-type Ⅱ functional response in emergency event, Physica A, 450 (2016), 228-240.  doi: 10.1016/j.physa.2015.12.143.

[10]

A. J. Kimmel and A.-F. Audrain-Pontevia, Analysis of commercial rumors from the perspective of marketing managers: Rumor prevalence, effects, and control tactics, Journal of Marketing Communications, 16 (2010), 239-253.  doi: 10.1080/13527260902884433.

[11]

D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.

[12]

M. McDonald, O. Suleman, S. Williams, S. Howison and N. F. Johnson, Impact of unexpected events, shocking news, and rumors on foreign exchange market dynamics, Physical Review E, 77 (2008), 046110. doi: 10.1103/PhysRevE.77.046110.

[13]

Y. Moreno, M. Nekovee and A. Pacheco, Dynamics of rumor spreading in complex networks, Physical Review E, 69 (2004), 066130. doi: 10.1103/PhysRevE.69.066130.

[14]

M. Nagao, K. Suto and A. Ohuchi, A media information analysis for implementing effective countermeasure against harmful rumor, Journal of Physics, Conference Series, 221 (2010), 012004. doi: 10.1088/1742-6596/221/1/012004.

[15]

M. Nekovee, Y. Moreno, G. Bianconi and M. Marsili, Theory of rumour spreading in complex social networks, Physica A, 374 (2007), 457-470.  doi: 10.1016/j.physa.2006.07.017.

[16]

R. L. Rosnow, J. H. Yost and J. L. Esposito, Belief in rumor and likelihood of rumor transmission, Language & Communication, 6 (1986), 189-194.  doi: 10.1016/0271-5309(86)90022-4.

[17]

A. Sudbury, The proportion of population never hearing a rumour, Journal of Applied Probability, 22 (1985), 443-446.  doi: 10.2307/3213787.

[18]

S. A. Thomas, Lies, damn lies, and rumors: An analysis of collective efficacy, rumors, and fear in the wake of Katrina, Sociological Spectrum, 27 (2007), 679-703.  doi: 10.1080/02732170701534200.

[19]

Y.-Q. Wang, X.-Y. Yang, Y.-L. Han and X.-A. Wang, Rumor Spreading Model with Trust Mechanism in Complex Social Networks, Communications in Theoretical Physics, 59 (2013), 510-516. 

[20]

J. Wang, L. Zhao and R. Huang, 2SI2R rumor spreading model in homogeneous networks, Physica A, 413 (2014), 153-161.  doi: 10.1016/j.physa.2014.06.053.

[21]

D. J. Watts and S. H. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), 440-442. 

[22]

Y. Zan, J. Wua, P. Li and Q. Yua, SICR rumor spreading model in complex networks: Counterattack and self-resistance, Physica A, 405 (2014), 159-170.  doi: 10.1016/j.physa.2014.03.021.

[23]

D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901. doi: 10.1103/PhysRevE.64.050901.

[24]

D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908. doi: 10.1103/PhysRevE.65.041908.

[25]

L. Zhao, H. Cui, X. Qiu, X. Wang and J. Wang, SIR rumor spreading model in the new media age, Physica A, 392 (2013), 995-1003.  doi: 10.1016/j.physa.2012.09.030.

[26]

L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng and H. Cui, SIHR rumor spreading model in social networks, Physica A, 391 (2012), 2444-2453. 

[27]

L. Zhao, Q. Wang, J. Cheng, Y. Chen, J. Wang and W. Huang, Rumor spreading model with consideration of forgetting mechanism: A case of online blogging LiveJournal, Physica A, 390 (2011), 2619-2625.  doi: 10.1016/j.physa.2011.03.010.

[28]

L. Zhao, W. Xie, H. O. Gao, X. Qiu, X. Wang and S. Zhang, A rumor spreading model with variable forgetting rate, Physica A, 392 (2013), 6146-6154.  doi: 10.1016/j.physa.2013.07.080.

[29]

L. Zhu, H. Zhao and H. Wang, Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms, Information Sciences, 349/350 (2016), 119-136.  doi: 10.1016/j.ins.2016.02.031.

Figure 1.  Temporal evolution of $\phi(t)$ with respect to $\delta$ and the plot of $F(\phi)$
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Figure 2.  Temporal evolution of the densities $S(t)$ and $R(t)$ with respect to $\delta$
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Figure 3.  The final size of the rumor $\phi^\infty$ with respect to $\delta$
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