American Institute of Mathematical Sciences

Advanced
April  2020, 25(4): 1497-1515. doi: 10.3934/dcdsb.2019237

An SICR rumor spreading model in heterogeneous networks

Jinxian Li 1,2,, , Ning Ren 1, and  Zhen Jin 2,3,

1. 

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
2. 

Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Taiyuan 030006, China
3. 

Complex System Research Center, Shanxi University, Taiyuan 030006, China

* Corresponding author: lijinxian@sxu.edu.cn

Received  December 2018 Revised  July 2019 Published  April 2020 Early access  November 2019

This article discusses the spread of rumors in heterogeneous networks. Using the probability generating function method and the approximation theory, we establish an SICR rumor model and calculate the threshold conditions for the outbreak of the rumor. We also compare the speed of the rumors spreading with different initial conditions. The numerical simulations of the SICR model in this paper fit well with the stochastic simulations, which means that the model is reliable. Moreover the effects of the parameters in the model on the transmission of rumors are studied numerically.

Keywords: Rumor spreading, SICR model, heterogeneous networks, counterattack, probability generating function.
Mathematics Subject Classification: Primary: 05C80, 37N25; Secondary: 92D25.
Citation: Jinxian Li, Ning Ren, Zhen Jin. An SICR rumor spreading model in heterogeneous networks. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1497-1515. doi: 10.3934/dcdsb.2019237
References:
[1]

L. Buzna, K. Peters and D. Helbing, Modelling the dynamics of disaster spreading in networks, Physica A, 363 (2006), 132-140.   Google Scholar

[2]

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.  Google Scholar

[3]

L.-A. Huo, P. Q. Huang and X. Fang, An interplay model for authorities' actions and rumor spreading in emergency event, Physica A, 390 (2011), 3267-3274.  doi: 10.1016/j.physa.2011.05.008.  Google Scholar

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L. A. Huo, L. Wang, N. X. Song, C. Y. Ma and B. He, Rumor spreading model considering the activity of spreaders in the homogeneous network, Physica A, 468 (2017), 855-865.  doi: 10.1016/j.physa.2016.11.039.  Google Scholar

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R. L. Jie, J. Qiao, G. J. Xu and Y. Y. Meng, A study on the interaction between two rumors in homogeneous complex networks under symmetric conditions, Physica A, 454 (2016), 129-142.  doi: 10.1016/j.physa.2016.02.048.  Google Scholar

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C. Lefévre and P. Picard, Distribution of the final extent of a rumour process, Journal of Applied Probability, 31 (1994), 244-249.  doi: 10.2307/3215250.  Google Scholar

[7]

J. X. Li, J. Wang and Z. Jin, SIR dynamics in random networks with communities, Journal of Mathematical Biology, 77 (2018), 1117-1151.  doi: 10.1007/s00285-018-1247-5.  Google Scholar

[8]

Q. M. Liu, T. Li and M. C. Su, The analysis of an SEIR rumor propagation model on heterogeneous network, Physica A, 469 (2017), 372-380.  doi: 10.1016/j.physa.2016.11.067.  Google Scholar

[9]

K. Ma, W. H. Li, Q. T. Guo, X. Q. Zheng, Z. M. Zheng, C. Gao and S. T. Tang, Information spreading in complex networks with participation of independent spreaders, Physica A, 492 (2018), 21-27.  doi: 10.1016/j.physa.2017.09.052.  Google Scholar

[10]

H. Matsuda, N. Ogita, A. Sasaki and K. Sato, Statistical mechanics of population: The lattice Lotka-Volterra model, Progress of Theoretical Physics, 88 (1992), 1035-1049.  doi: 10.1143/ptp/88.6.1035.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

M. E. J. Newman, S. Forrest and J. Balthrop, Email networks and the spread of computer viruses, Physical Review E, 66 (2002), 035101. doi: 10.1103/PhysRevE.66.035101.  Google Scholar

[14]

Z. Qian, S. T. Tang, X. Zhang and Z. M. Zheng, The independent spreaders involved SIR Rumor model in complex networks, Physica A, 429 (2015), 95-102.  doi: 10.1016/j.physa.2015.02.022.  Google Scholar

[15]

F. Roshani and Y. Naimi, Effects of degree-biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks, Physical Review E, 85 (2012), 036109. Google Scholar

[16]

Z. Y. Ruan, M. Tang and Z. H. Liu, Epidemic spreading with information-driven vaccination, Physical Review E, 86 (2012), 036117. doi: 10.1103/PhysRevE.86.036117.  Google Scholar

[17]

E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293-310.  doi: 10.1007/s00285-007-0116-4.  Google Scholar

[18]

C. Wan, T. Li and Y. Wang, Rumor Spreading of a SICS Model on complex social networks with counter mechanism, Open Access Library Journal, 03 (2016), 1-11.   Google Scholar

[19]

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

[20]

J. J. Wang, L. J. Zhao and R. B. Huang, SIRaRu rumor spreading model in complex networks, Physica A, 398 (2014), 43-55.  doi: 10.1016/j.physa.2013.12.004.  Google Scholar

[21]

Y. Wang, X. Yang and Y. Han, Rumor spreading model with trust mechanism in complex social networks, Communications in Theoretical Physics, 59 (2013), 510-516.   Google Scholar

[22]

L.-L. Xia, G.-P. Jiang, B. Song and Y.-R. Song, Rumor spreading model considering hesitating mechanism in complex social networks, Physica A, 437 (2015), 295-303.  doi: 10.1016/j.physa.2015.05.113.  Google Scholar

[23]

Y. L. Zan, J. L. Wu, P. Li and Q. L. Yu, 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.  Google Scholar

[24]

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

[25]

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

[26]

L. J. Zhao, X. Y. Qiu, X. L. Wang and J. J. Wang, Rumor spreading model considering forgetting and remembering mechanisms in inhomogeneous networks, Physica A, 392 (2013), 987-994.  doi: 10.1016/j.physa.2012.10.031.  Google Scholar

[27]

L. J. Zhao, J. J. Wang, Y. C. Chen, Q. Wang, J. J. Cheng and H. X. Cui, SIHR rumor spreading model in social networks, Physica A, 391 (2012), 2444-2453.  doi: 10.1016/j.physa.2011.12.008.  Google Scholar

[28]

L. Zhu and Y. G. Wang, Rumor spreading model with noise interference in complex social networks, Physica A, 469 (2017), 750-760.  doi: 10.1016/j.physa.2016.11.119.  Google Scholar

show all references

References:
[1]

L. Buzna, K. Peters and D. Helbing, Modelling the dynamics of disaster spreading in networks, Physica A, 363 (2006), 132-140.   Google Scholar

[2]

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.  Google Scholar

[3]

L.-A. Huo, P. Q. Huang and X. Fang, An interplay model for authorities' actions and rumor spreading in emergency event, Physica A, 390 (2011), 3267-3274.  doi: 10.1016/j.physa.2011.05.008.  Google Scholar

[4]

L. A. Huo, L. Wang, N. X. Song, C. Y. Ma and B. He, Rumor spreading model considering the activity of spreaders in the homogeneous network, Physica A, 468 (2017), 855-865.  doi: 10.1016/j.physa.2016.11.039.  Google Scholar

[5]

R. L. Jie, J. Qiao, G. J. Xu and Y. Y. Meng, A study on the interaction between two rumors in homogeneous complex networks under symmetric conditions, Physica A, 454 (2016), 129-142.  doi: 10.1016/j.physa.2016.02.048.  Google Scholar

[6]

C. Lefévre and P. Picard, Distribution of the final extent of a rumour process, Journal of Applied Probability, 31 (1994), 244-249.  doi: 10.2307/3215250.  Google Scholar

[7]

J. X. Li, J. Wang and Z. Jin, SIR dynamics in random networks with communities, Journal of Mathematical Biology, 77 (2018), 1117-1151.  doi: 10.1007/s00285-018-1247-5.  Google Scholar

[8]

Q. M. Liu, T. Li and M. C. Su, The analysis of an SEIR rumor propagation model on heterogeneous network, Physica A, 469 (2017), 372-380.  doi: 10.1016/j.physa.2016.11.067.  Google Scholar

[9]

K. Ma, W. H. Li, Q. T. Guo, X. Q. Zheng, Z. M. Zheng, C. Gao and S. T. Tang, Information spreading in complex networks with participation of independent spreaders, Physica A, 492 (2018), 21-27.  doi: 10.1016/j.physa.2017.09.052.  Google Scholar

[10]

H. Matsuda, N. Ogita, A. Sasaki and K. Sato, Statistical mechanics of population: The lattice Lotka-Volterra model, Progress of Theoretical Physics, 88 (1992), 1035-1049.  doi: 10.1143/ptp/88.6.1035.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

M. E. J. Newman, S. Forrest and J. Balthrop, Email networks and the spread of computer viruses, Physical Review E, 66 (2002), 035101. doi: 10.1103/PhysRevE.66.035101.  Google Scholar

[14]

Z. Qian, S. T. Tang, X. Zhang and Z. M. Zheng, The independent spreaders involved SIR Rumor model in complex networks, Physica A, 429 (2015), 95-102.  doi: 10.1016/j.physa.2015.02.022.  Google Scholar

[15]

F. Roshani and Y. Naimi, Effects of degree-biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks, Physical Review E, 85 (2012), 036109. Google Scholar

[16]

Z. Y. Ruan, M. Tang and Z. H. Liu, Epidemic spreading with information-driven vaccination, Physical Review E, 86 (2012), 036117. doi: 10.1103/PhysRevE.86.036117.  Google Scholar

[17]

E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293-310.  doi: 10.1007/s00285-007-0116-4.  Google Scholar

[18]

C. Wan, T. Li and Y. Wang, Rumor Spreading of a SICS Model on complex social networks with counter mechanism, Open Access Library Journal, 03 (2016), 1-11.   Google Scholar

[19]

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

[20]

J. J. Wang, L. J. Zhao and R. B. Huang, SIRaRu rumor spreading model in complex networks, Physica A, 398 (2014), 43-55.  doi: 10.1016/j.physa.2013.12.004.  Google Scholar

[21]

Y. Wang, X. Yang and Y. Han, Rumor spreading model with trust mechanism in complex social networks, Communications in Theoretical Physics, 59 (2013), 510-516.   Google Scholar

[22]

L.-L. Xia, G.-P. Jiang, B. Song and Y.-R. Song, Rumor spreading model considering hesitating mechanism in complex social networks, Physica A, 437 (2015), 295-303.  doi: 10.1016/j.physa.2015.05.113.  Google Scholar

[23]

Y. L. Zan, J. L. Wu, P. Li and Q. L. Yu, 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.  Google Scholar

[24]

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

[25]

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

[26]

L. J. Zhao, X. Y. Qiu, X. L. Wang and J. J. Wang, Rumor spreading model considering forgetting and remembering mechanisms in inhomogeneous networks, Physica A, 392 (2013), 987-994.  doi: 10.1016/j.physa.2012.10.031.  Google Scholar

[27]

L. J. Zhao, J. J. Wang, Y. C. Chen, Q. Wang, J. J. Cheng and H. X. Cui, SIHR rumor spreading model in social networks, Physica A, 391 (2012), 2444-2453.  doi: 10.1016/j.physa.2011.12.008.  Google Scholar

[28]

L. Zhu and Y. G. Wang, Rumor spreading model with noise interference in complex social networks, Physica A, 469 (2017), 750-760.  doi: 10.1016/j.physa.2016.11.119.  Google Scholar

Figure 1.  Structure of the rumor spreading process
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23]. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size">Figure 2.  Simulation with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $. The blue dotted lines correspond to 100 random simulations for an SICR rumor model in a network with the Poisson degree distribution. The red solid lines show the numerical simulation based on the model (1). The green solid lines show the numerical simulation based on the mean field system in Zan [23]. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size
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Figure 3.  Degree distribution of generated Power-law networks
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23]. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size">Figure 4.  Simulations with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $. The blue dotted lines correspond to 100 random simulations for an SICR rumor model in a network with the power law degree destribution. The red solid lines show the numerical simulation based on the model (1). The green solid lines show the numerical simulation based on the mean field system in Zan [23]. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size
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Figure 5.  Numerical simulations of the model (1) with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $ in the networks with different degree distribution but with the same averaged degree. The red dotted lines and the blue solid lines correspond to numerical simulations trajectories based on the model (1) in a network with the refined power-law degree distribution and the Poisson degree distribution, respectively. (a) the trajectories of the densities of the infective. (b) the trajectories of the densities of the final size
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Figure 6.  (a) Time evolutions of the speeds of the rumor spreading $ v $ in the complex network with poisson degree distribution and Power-law distribution, respectively. (b) Degree distriutions
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Figure 7.  Time evolutions of the speeds of the rumor spreading in the nodes with degree $ k = 5, 10, 15. $, i.e. $ v_k $. (a) In network with Poisson ditribution. (b) In network with Power Law distribution
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Figure 8.  Time evolutions of the relative propagation velocity in the nodes with degree $ k = 5, 10, 15. $, i.e. $ \hat{v}_k $. (a) In network with Poisson ditribution. (b) In network with Power Law distribution
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Figure 9.  numerical simulations with $ \alpha = 0.7 $, $ \beta = 0.1 $, $ \delta = 0.2 $, $ \gamma = 0.2 $. (a) the trajectories of the densities of the infective over time under different persuading rate $ \eta $. (b) the trajectories of the densities of the final size over time under different persuading rate $ \eta $
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Figure 10.  Numerical simulations with $ \beta = 0.2 $, $ \gamma = 0.5 $, $ \eta = 0.8 $. (a) the trajectories of the densities of the susceptible over time under different spreading rate $ \alpha $ and refuting rate $ \delta $. (b) the trajectories of the densities of the infective over time under different spreading rate $ \alpha $ and refuting rate $ \delta $. (c) the trajectories of the densities of the final size over time under different spreading rate $ \alpha $ and refuting rate $ \delta $
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Figure 11.  Numerical simulation of SIR model that compare the infective nodes recover by themselves and recover by the structure of the network. (a) the trajectories of the densities of the susceptible. (b) the trajectories of the densities of the infective. (c) the trajectories of the densities of the final size
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Table 1.  Key variables and parameters
Series Symbol Series Description
$ \alpha $ Spreading rate. The constant rate at which a susceptible node
becomes an infective node when it contacts an infective node
$ \beta $ Ignoring rate. The constant rate at which a susceptible node becomes
a refractory node when it contacts an infective node
$ \delta $ Refuting rate. The constant rate at which a susceptible node becomes
a counterattack node when it contacts an infective node
$ \gamma $ Stifling rate. The constant rate at which an infective node becomes
a refractory node when it contacts another infective or
refractory node
$ \eta $ Persuading rate. The constant rate at which an infective node
becomes a refractory node when it contacts a counterattack node
$ p_k $ The probability that a node will have degree $ k $
$ g(x) $ The probability generating function for the degree distribution $ \{p_k\} $
$ P_X^Y $ The probability that an arc with an ego in set X has an alter in Y
$ \mathcal{A}_X $ Set of arcs (ego, alter) such that node ego is in set $ X $
$ M_X $ Fraction of arcs in set $ \mathcal{A}_X $
$ \mathcal{A}_{XY} $ Set of arcs (ego, alter) s.t ego$ \in X $ and alter$ \in Y $
$ M_{XY} $ Fraction of arcs in set $ \mathcal{A}_{XY} $
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Series Symbol Series Description
$ \alpha $ Spreading rate. The constant rate at which a susceptible node
becomes an infective node when it contacts an infective node
$ \beta $ Ignoring rate. The constant rate at which a susceptible node becomes
a refractory node when it contacts an infective node
$ \delta $ Refuting rate. The constant rate at which a susceptible node becomes
a counterattack node when it contacts an infective node
$ \gamma $ Stifling rate. The constant rate at which an infective node becomes
a refractory node when it contacts another infective or
refractory node
$ \eta $ Persuading rate. The constant rate at which an infective node
becomes a refractory node when it contacts a counterattack node
$ p_k $ The probability that a node will have degree $ k $
$ g(x) $ The probability generating function for the degree distribution $ \{p_k\} $
$ P_X^Y $ The probability that an arc with an ego in set X has an alter in Y
$ \mathcal{A}_X $ Set of arcs (ego, alter) such that node ego is in set $ X $
$ M_X $ Fraction of arcs in set $ \mathcal{A}_X $
$ \mathcal{A}_{XY} $ Set of arcs (ego, alter) s.t ego$ \in X $ and alter$ \in Y $
$ M_{XY} $ Fraction of arcs in set $ \mathcal{A}_{XY} $
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