# American Institute of Mathematical Sciences

April  2019, 24(4): 1843-1865. doi: 10.3934/dcdsb.2018240

## Spreading-vanishing dichotomy in information diffusion in online social networks with intervention

 1 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China 2 School of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069, USA

* Corresponding author: Jingli Ren

Received  November 2017 Revised  April 2018 Published  August 2018

In this paper, multiple information diffusion in online social networks with free boundary condition is investigated. We prove a spreading-vanishing dichotomy for the problem: i.e., either at least one piece of information lasts forever or all information suspend in finite time. The criterion for spreading and vanishing is established, it is related to the initial spreading area and the expansion capacity. We also obtain several cases of the asymptotic behavior of the information under different conditions. When the information spreads, we provide some upper and lower bounds of the spreading speed corresponding to different cases of asymptotic behavior of the information. In addition, numerical examples are given to illustrate the impacts of the initial spreading area and the expansion capacity on the free boundary, and all cases of the asymptotic behavior of the information.

Citation: Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240
##### References:
 [1] I. Ahn, S. Baek and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model., 40 (2016), 7082-7101. doi: 10.1016/j.apm.2016.02.038. Google Scholar [2] F. Benvenuto, T. Rodrigues, M. Cha and V. Almeida, Characterizing user behavior in online social networks, in 9th ACM SIGCOMM Internet Measurement Conference, (2009), 49-62. doi: 10.1145/1644893.1644900. Google Scholar [3] G. Bunting, Y. H. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583. Google Scholar [4] R. S. Cantrell and C. Consner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd., Chichester, 2003. doi: 10.1002/0470871296. Google Scholar [5] X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. Google Scholar [6] Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. Google Scholar [7] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar [8] Y. H. Du and L. Ma, Logistic type equations on $\mathbb{R}^{N}$ by a squeezing method involving boundary blow-up solutions, J. Lond. Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289. Google Scholar [9] R. Ghosh and K. Lerman, A framework for quantitative analysis of cascades on networks, WSDM '11 Proceedings of the fourth ACM international conference on Web search and data mining, (2011), 665-674. doi: 10.1145/1935826.1935917. Google Scholar [10] J. S. Guo and C. H. Wu, On a free boundary for a two-species weak competition system, J. Dynam. Diff. Equat., 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0. Google Scholar [11] J. Jiang, C. Wilson, X. Wang, P. Huang, W. P. Sha, Y. F. Dai and B. Y. Zhao, Understanding latent interactions in online social networks, in Proceedings of ACM SIGCOMM International Measurement Conference, (2010), 369-382. doi: 10.1145/1879141.1879190. Google Scholar [12] A. Kolmogorov, I. Petrovski and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech., 1 (1937), 1-25. Google Scholar [13] K. Lerman and R. Ghosh, Information contagion: An empirical study of spread of news on digg and twitter social networks, in Proceedings of International Conference on Weblogs and Social Media (ICWSM), 2010.Google Scholar [14] C. X. Lei, Z. G. Lin and H. Y. Wang, The free bondary problem describing information diffusion in online social networks, J. Differential Equations, 254 (2013), 1326-1341. doi: 10.1016/j.jde.2012.10.021. Google Scholar [15] C. X. Lei, Z. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015. Google Scholar [16] G. Lin, W. T. Li and M. J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. Google Scholar [17] Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409. doi: 10.1007/s00285-017-1124-7. Google Scholar [18] J. D. Murray and R. P. Sperb, Minimum domains for spatial patterns in a class of reaction diffusion equations, J. Math. Biol., 18 (1983), 169-184. doi: 10.1007/BF00280665. Google Scholar [19] C. Peng, K. Xu, F. Wang and H. Y. Wang, Predicting information diffusion initiated from multiple sources in online social networks, in 6th International Symposium on Computational Intelligence and Design(ISCID), (2013), 96-99. doi: 10.1109/ISCID.2013.138. Google Scholar [20] S. Razvan and D. Gabriel, Numerical approximation of a free boundary problem for a predator-prey model, Numer. Anal. Appl., 5434 (2009), 548-555. Google Scholar [21] J. L. Ren and L. P. Yu, Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model, J. Nonlinear Sci., 26 (2016), 1895-1931. doi: 10.1007/s00332-016-9323-8. Google Scholar [22] L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971. Google Scholar [23] G. V. Steeg, R. Ghosh and K. Lerman, What stops social epidemics? in ICWSM '11: Proceedings of the 5th Int. Conf. on Weblogs and Social Media, 2011.Google Scholar [24] F. Wang, H. Y. Wang and K. Xu, Diffusive logistic model towards predicting information diffusion in online social networks, in 32nd International Conference on Distributed Computing Systems Workshops (ICDCS), (2012), 133-139. doi: 10.1109/ICDCSW.2012.16. Google Scholar [25] F. Wang, H. Y. Wang and K. Xu, Characterizing information diffusion in online social networks with linear diffusive model, in 33rd IEEE International Conference on Distributed Computing Systems (ICDCS), (2013), 307-316. doi: 10.1109/ICDCS.2013.14. Google Scholar [26] M. X. Wang and J. F. Zhao, Free boundary problems for the Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4. Google Scholar [27] M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013. Google Scholar [28] M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat., 29 (2017), 957-979. doi: 10.1007/s10884-015-9503-5. Google Scholar [29] Y. Xu, D. D. Zhu and J. L. Ren, On a reaction-diffusion-advection system: Fixed boundary vs free boundary, Electron. J. Qual. Theod., (2018), in press.Google Scholar [30] J. Yang and J. Leskovec, Modeling information diffusion in implicit networks, in Proceedings of IEEE International Conference on Data Mining, 2010. doi: 10.1109/ICDM.2010.22. Google Scholar [31] S. Z. Ye and S. F. Wu, Measuring message propagation and social influence on Twitter.com, International Conference on Social Informatics, (2010), 216-231. doi: 10.1007/978-3-642-16567-2_16. Google Scholar [32] D. D. Zhu, J. L. Ren and H. P. Zhu, Spatial-temporal basic reproduction number and dynamics for a dengue disease diffusion model, Math. Meth. Appl. Sci., (2018), in press. doi: 10.1002/mma.5085. Google Scholar [33] L. H. Zhu, H. Y. Zhao and H. Y. Wang, Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms, Inform. Sci., 349/350 (2016), 119-136. doi: 10.1016/j.ins.2016.02.031. Google Scholar

show all references

##### References:
 [1] I. Ahn, S. Baek and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model., 40 (2016), 7082-7101. doi: 10.1016/j.apm.2016.02.038. Google Scholar [2] F. Benvenuto, T. Rodrigues, M. Cha and V. Almeida, Characterizing user behavior in online social networks, in 9th ACM SIGCOMM Internet Measurement Conference, (2009), 49-62. doi: 10.1145/1644893.1644900. Google Scholar [3] G. Bunting, Y. H. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583. Google Scholar [4] R. S. Cantrell and C. Consner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd., Chichester, 2003. doi: 10.1002/0470871296. Google Scholar [5] X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693. Google Scholar [6] Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011. Google Scholar [7] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089. Google Scholar [8] Y. H. Du and L. Ma, Logistic type equations on $\mathbb{R}^{N}$ by a squeezing method involving boundary blow-up solutions, J. Lond. Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289. Google Scholar [9] R. Ghosh and K. Lerman, A framework for quantitative analysis of cascades on networks, WSDM '11 Proceedings of the fourth ACM international conference on Web search and data mining, (2011), 665-674. doi: 10.1145/1935826.1935917. Google Scholar [10] J. S. Guo and C. H. Wu, On a free boundary for a two-species weak competition system, J. Dynam. Diff. Equat., 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0. Google Scholar [11] J. Jiang, C. Wilson, X. Wang, P. Huang, W. P. Sha, Y. F. Dai and B. Y. Zhao, Understanding latent interactions in online social networks, in Proceedings of ACM SIGCOMM International Measurement Conference, (2010), 369-382. doi: 10.1145/1879141.1879190. Google Scholar [12] A. Kolmogorov, I. Petrovski and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ. Math. Mech., 1 (1937), 1-25. Google Scholar [13] K. Lerman and R. Ghosh, Information contagion: An empirical study of spread of news on digg and twitter social networks, in Proceedings of International Conference on Weblogs and Social Media (ICWSM), 2010.Google Scholar [14] C. X. Lei, Z. G. Lin and H. Y. Wang, The free bondary problem describing information diffusion in online social networks, J. Differential Equations, 254 (2013), 1326-1341. doi: 10.1016/j.jde.2012.10.021. Google Scholar [15] C. X. Lei, Z. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166. doi: 10.1016/j.jde.2014.03.015. Google Scholar [16] G. Lin, W. T. Li and M. J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414. doi: 10.3934/dcdsb.2010.13.393. Google Scholar [17] Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409. doi: 10.1007/s00285-017-1124-7. Google Scholar [18] J. D. Murray and R. P. Sperb, Minimum domains for spatial patterns in a class of reaction diffusion equations, J. Math. Biol., 18 (1983), 169-184. doi: 10.1007/BF00280665. Google Scholar [19] C. Peng, K. Xu, F. Wang and H. Y. Wang, Predicting information diffusion initiated from multiple sources in online social networks, in 6th International Symposium on Computational Intelligence and Design(ISCID), (2013), 96-99. doi: 10.1109/ISCID.2013.138. Google Scholar [20] S. Razvan and D. Gabriel, Numerical approximation of a free boundary problem for a predator-prey model, Numer. Anal. Appl., 5434 (2009), 548-555. Google Scholar [21] J. L. Ren and L. P. Yu, Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model, J. Nonlinear Sci., 26 (2016), 1895-1931. doi: 10.1007/s00332-016-9323-8. Google Scholar [22] L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971. Google Scholar [23] G. V. Steeg, R. Ghosh and K. Lerman, What stops social epidemics? in ICWSM '11: Proceedings of the 5th Int. Conf. on Weblogs and Social Media, 2011.Google Scholar [24] F. Wang, H. Y. Wang and K. Xu, Diffusive logistic model towards predicting information diffusion in online social networks, in 32nd International Conference on Distributed Computing Systems Workshops (ICDCS), (2012), 133-139. doi: 10.1109/ICDCSW.2012.16. Google Scholar [25] F. Wang, H. Y. Wang and K. Xu, Characterizing information diffusion in online social networks with linear diffusive model, in 33rd IEEE International Conference on Distributed Computing Systems (ICDCS), (2013), 307-316. doi: 10.1109/ICDCS.2013.14. Google Scholar [26] M. X. Wang and J. F. Zhao, Free boundary problems for the Lotka-Volterra competition system, J. Dyn. Diff. Equat., 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4. Google Scholar [27] M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013. Google Scholar [28] M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dyn. Diff. Equat., 29 (2017), 957-979. doi: 10.1007/s10884-015-9503-5. Google Scholar [29] Y. Xu, D. D. Zhu and J. L. Ren, On a reaction-diffusion-advection system: Fixed boundary vs free boundary, Electron. J. Qual. Theod., (2018), in press.Google Scholar [30] J. Yang and J. Leskovec, Modeling information diffusion in implicit networks, in Proceedings of IEEE International Conference on Data Mining, 2010. doi: 10.1109/ICDM.2010.22. Google Scholar [31] S. Z. Ye and S. F. Wu, Measuring message propagation and social influence on Twitter.com, International Conference on Social Informatics, (2010), 216-231. doi: 10.1007/978-3-642-16567-2_16. Google Scholar [32] D. D. Zhu, J. L. Ren and H. P. Zhu, Spatial-temporal basic reproduction number and dynamics for a dengue disease diffusion model, Math. Meth. Appl. Sci., (2018), in press. doi: 10.1002/mma.5085. Google Scholar [33] L. H. Zhu, H. Y. Zhao and H. Y. Wang, Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms, Inform. Sci., 349/350 (2016), 119-136. doi: 10.1016/j.ins.2016.02.031. Google Scholar
The relationship among three information
$u, v$ and $w$ all vanish
$u, v$ and $w$ all spread
$u, v$ and $w$ all spread
$u, v$ and $w$ all spread
$u$ and $v$ vanish, $w$ spreads
$u$ vanishes, $v$ and $w$ spread
$v$ vanishes, $u$ and $w$ spread
$u, v$ and $w$ all spread
The density of influenced users of information A varies with the increase of the intervention rate $c_{1}$ for (A) and with the increase of the competition rate $b_{1}$ for (B)
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