A physics-informed neural network model for social media user growth

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, a physics-informed neural network model is proposed to predict the growth of online social network users. The number of online social network users is modeled by a stochastic process and the associated Kolmogorov forward equation is derived. Then, a physics-informed neural network model is built based on the Kolmogorov forward equation and trained using real-world data. By combining mathematical modeling with machine learning, our approach provides a practical and interpretable methodology that harnesses the strengths of both physical laws and advancements in machine learning, while minimizing the opacity in machine learning models.

Keywords

physics-informed neural network / Kolmogorov forward equations / online social networks / compartment models / Markov process

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] E. Allen, Modeling with Itô stochastic differential equations, Dordrecht: Springer, 2007. http://dx.doi.org/10.1007/978-1-4020-5953-7

[2] T. Boyle, R. Aygun, Kennesaw State University HPC facilities and resources, Digital Commons Training Materials, 10 (2021), 1-3.

[3] F. Brauer, Mathematical epidemiology: past, present, and future, Infectious Disease Modelling, 2 (2017), 113-127. http://dx.doi.org/10.1016/j.idm.2017.02.001

[4] C. Browne, M. Wang, G. F. Webb, A stochastic model of nosocomial epidemics in hospital intensive care units, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1-12. http://dx.doi.org/10.14232/ejqtde.2017.1.6

[5] J. Cannarella, J. Spechler, Epidemiological modeling of online network dynamics, arXiv: 1401.4208. http://dx.doi.org/10.48550/arXiv.1401.4208

[6] R. Chen, L. Kong, M. Wang, Stability analysis of an online social network model, Rocky Mountain J. Math., 53 (2023), 1019-1041. http://dx.doi.org/10.1216/rmj.2023.53.1019

[7] S. Chen, J. Shi, Z. Shuai, Y. Wu, Evolution of dispersal in advective patchy environments, J. Nonlinear Sci., 33 (2023), 40. http://dx.doi.org/10.1007/s00332-023-09899-w

[8] S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, F. Piccialli, Scientific machine learning through physics-informed neural networks: where we are and what’s next, J. Sci. Comput., 92 (2022), 88. http://dx.doi.org/10.1007/s10915-022-01939-z

[9] D. Gao, X. Yuan, A hybrid Lagrangian-Eulerian model for vector-borne diseases, J. Math. Biol., 89 (2024), 16. http://dx.doi.org/10.1007/s00285-024-02109-5

[10] J. R. Graef, S. Ho, L. Kong, M. Wang, A fractional differential equation model for bike share systems, J. Nonlinear Funct. Anal., 2019 (2019), 23. http://dx.doi.org/10.23952/jnfa.2019.23

[11] J. R. Graef, L. Kong, A. Ledoan, M. Wang, Stability analysis of a fractional online social network model, Math. Comput. Simulat., 178 (2020), 625-645. http://dx.doi.org/10.1016/j.matcom.2020.07.012

[12] Z. Hao, S. Liu, Y. Zhang, C. Ying, Y. Feng, H. Su, et al., Physics-informed machine learning: a survey on problems, methods and applications, arXiv: 2211.08064. http://dx.doi.org/10.48550/arXiv.2211.08064

[13] D. P. Kingma, J. Ba, Adam: a method for stochastic optimization, arXiv: 1412.6980. http://dx.doi.org/10.48550/arXiv.1412.6980

[14] N. Kimmel, L. Kong, M. Wang, Modeling the dynamics of user adoption and abandonment in online social networks, Math. Method. Appl. Sci., in press. http://dx.doi.org/10.1002/mma.10413

[15] D. Kincaid, W. Cheney, Numerical analysis: mathematics of scientific computing, 3 Eds., Providence: American Mathematical Society, 2002.

[16] L. Kong, Modelling the dynamics of product adoption and abandonment, Proc. R. Soc. A., 480 (2024), 20240034. http://dx.doi.org/10.1098/rspa.2024.0034

[17] L. Kong, M. Wang, Deterministic and stochastic online social network models with varying population size, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 30 (2023), 253-275.

[18] L. Kong, M. Wang, Optimal control for an ordinary differential equation online social network model, Differ. Equat. Appl., 14 (2022), 205-214.

[19] B. Ma, C. Li, J. Warner, Structured mathematical models to investigate the interactions between Plasmodium falciparum malaria parasites and host immune response, Math. Biosci., 310 (2019), 65-75. http://dx.doi.org/10.1016/j.mbs.2019.02.005

[20] M. Mohsin, 10 social media statistics you need to know in 2024, Oberlo, 2024. Available from: https://www.oberlo.com/blog/social-media-marketing-statistics.

[21] K. Nath, X. Meng, D. J. Smith, G. Karniadakis, Physics-informed neural networks for predicting gas flow dynamics and unknown parameters in diesel engines, Sci. Rep., 13 (2023), 13683. http://dx.doi.org/10.1038/s41598-023-39989-4

[22] E. Ortiz-Ospina, The rise of social media, Our World In Data, 2019. Available from: https://ourworldindata.org/rise-of-social-media.

[23] L. Wang, M. Wang, Stability and bifurcation analysis for an OSN model with delay, Advances in the Theory of Nonlinear Analysis and its Application, 7 (2023), 413-427. http://dx.doi.org/10.31197/atnaa.1152602

[24] L. Wang, M. Wang, Bifurcation analysis for an OSN model with two delays, Mathematics, 12 (2024), 1321. http://dx.doi.org/10.3390/math12091321

[25] G. Webb, X. E. Zhao, An epidemic model with infection age and vaccination age structure, Infect. Dis. Rep., 16 (2024), 35-64. http://dx.doi.org/10.3390/idr16010004

[26] B. Wong, Top social media statistics and trends of 2024, Forbes Media LLC., 2024. Available from: https://www.forbes.com/advisor/business/social-media-statistics/.

[27] World bank open data, The World Bank Group, 2024. Available from: https://data.worldbank.org/indicator/SP.POP.TOTL.

[28] N. Xiao, H. Xu, A. Morani, A. Shokri, H. Mukalazi, Exploring local and global stability of COVID-19 through numerical schemes, Sci. Rep., 14 (2024), 7960. http://dx.doi.org/10.1038/s41598-024-56938-x