Figure 1.
Structure of the rumor spreading process
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April
2020, 25(4): 1497-1515.
doi: 10.3934/dcdsb.2019237
An SICR rumor spreading model in heterogeneous networks
| 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 |
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 and 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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [24] |
D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901.
doi: 10.1103/PhysRevE.64.050901. |
| [25] |
D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908.
doi: 10.1103/PhysRevE.65.041908. |
| [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. |
| [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. |
| [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. |
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [24] |
D. H. Zanette, Critical behavior of propagation on small-world networks, Physical Review E, 64 (2001), 050901.
doi: 10.1103/PhysRevE.64.050901. |
| [25] |
D. H. Zanette, Dynamics of rumor propagation on small-world networks, Physical Review E, 65 (2002), 041908.
doi: 10.1103/PhysRevE.65.041908. |
| [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. |
| [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. |
| [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. |
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
Figure 3.
Degree distribution of generated Power-law networks
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
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
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
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
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
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 $
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 $
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
Table 1.
Key variables and parameters
| Series Symbol | Series Description |
| Spreading rate. The constant rate at which a susceptible node | |
| becomes an infective node when it contacts an infective node | |
| Ignoring rate. The constant rate at which a susceptible node becomes | |
| a refractory node when it contacts an infective node | |
| Refuting rate. The constant rate at which a susceptible node becomes | |
| a counterattack node when it contacts an infective node | |
| Stifling rate. The constant rate at which an infective node becomes | |
| a refractory node when it contacts another infective or | |
| refractory node | |
| Persuading rate. The constant rate at which an infective node | |
| becomes a refractory node when it contacts a counterattack node | |
| The probability that a node will have degree |
|
| The probability generating function for the degree distribution |
|
| The probability that an arc with an ego in set X has an alter in Y | |
| Set of arcs (ego, alter) such that node ego is in set |
|
| Fraction of arcs in set |
|
| Set of arcs (ego, alter) s.t ego |
|
| Fraction of arcs in set |
| Series Symbol | Series Description |
| Spreading rate. The constant rate at which a susceptible node | |
| becomes an infective node when it contacts an infective node | |
| Ignoring rate. The constant rate at which a susceptible node becomes | |
| a refractory node when it contacts an infective node | |
| Refuting rate. The constant rate at which a susceptible node becomes | |
| a counterattack node when it contacts an infective node | |
| Stifling rate. The constant rate at which an infective node becomes | |
| a refractory node when it contacts another infective or | |
| refractory node | |
| Persuading rate. The constant rate at which an infective node | |
| becomes a refractory node when it contacts a counterattack node | |
| The probability that a node will have degree |
|
| The probability generating function for the degree distribution |
|
| The probability that an arc with an ego in set X has an alter in Y | |
| Set of arcs (ego, alter) such that node ego is in set |
|
| Fraction of arcs in set |
|
| Set of arcs (ego, alter) s.t ego |
|
| Fraction of arcs in set |
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