A model of an online social network (OSN)

Eric R. Kaufmann; Joie Lea Murorunkwere; Eric R. Kaufmann; Joie Lea Murorunkwere

doi:10.3934/era.2026282

Electronic Research Archive

2026, Volume 34, Issue 9: 6448-6462. doi: 10.3934/era.2026282

Previous Article Next Article

Research article Special Issues

A model of an online social network (OSN)

Eric R. Kaufmann ,
Joie Lea Murorunkwere ^,

Department of Mathematics & Statistics, University of Arkansas at Little Rock, 2801 S. University Avenue, Little Rock, AR 72204, USA

Received: 10 April 2026 Revised: 02 June 2026 Accepted: 24 June 2026 Published: 22 July 2026

We introduce a new epidemic framework to model the adoption and abandonment dynamics of users on an online social network (OSN), with special emphasis on the novel compartment pauci-engaged. Pauci-engaged individuals have been exposed to an invitation to an OSN, but remain undecided about joining, thus reflecting real-world hesitation and trial behaviors. Building on the infectious–recovery dynamics studied by Chen, Kong, and Wang, we derive the basic reproduction number $ \mathscr{R}_0 $ and establish threshold conditions for the stability of both the user-free equilibrium and the user-prevailing equilibrium. Local stability analysis using the Jacobian matrix, together with global stability results based on Lyapunov functions, shows that when the basic reproduction number $ \mathscr{R}_0 < 1 $, the OSN cannot sustain an engaged user base, while when $ \mathscr{R}_0 > 1 $, a persistent engagement state emerges. Our findings highlight the pivotal role of the decision-making phase in the growth of the digital platform and offer actionable insights to optimize user recruitment and retention strategies in OSNs.
- online social network,
- equilibrium,
- stability analysis
Citation: Eric R. Kaufmann, Joie Lea Murorunkwere. A model of an online social network (OSN)[J]. Electronic Research Archive, 2026, 34(9): 6448-6462. doi: 10.3934/era.2026282

Related Papers:

Abstract

We introduce a new epidemic framework to model the adoption and abandonment dynamics of users on an online social network (OSN), with special emphasis on the novel compartment pauci-engaged. Pauci-engaged individuals have been exposed to an invitation to an OSN, but remain undecided about joining, thus reflecting real-world hesitation and trial behaviors. Building on the infectious–recovery dynamics studied by Chen, Kong, and Wang, we derive the basic reproduction number $ \mathscr{R}_0 $ and establish threshold conditions for the stability of both the user-free equilibrium and the user-prevailing equilibrium. Local stability analysis using the Jacobian matrix, together with global stability results based on Lyapunov functions, shows that when the basic reproduction number $ \mathscr{R}_0 < 1 $, the OSN cannot sustain an engaged user base, while when $ \mathscr{R}_0 > 1 $, a persistent engagement state emerges. Our findings highlight the pivotal role of the decision-making phase in the growth of the digital platform and offer actionable insights to optimize user recruitment and retention strategies in OSNs.

References

[1]	M. Bailey, R. Cao, T. Kuchler, J. Stroebel, The economic effects of social networks: Evidence from the housing market, J. Polit. Econ., 126 (2018), 2224–2276. https://doi.org/10.1086/700073 doi: 10.1086/700073
[2]	D. M. Boyd, N. B. Ellison, Social networks sites: Definition, history and scholarship, J. Comput.-Mediated Commun., 13 (2007), 210–230. https://doi.org/10.1111/j.1083-6101.2007.00393.x doi: 10.1111/j.1083-6101.2007.00393.x
[3]	F. Griffiths, T. Dobermann, J. A. K. Cave, M. Thorogood, S. Johnson, K. Salamatian, et al., The impact of online social networks on health and health systems: A scoping review and case studies, Policy Internet, 7 (2015), 473–496. https://doi.org/10.1002/poi3.97 doi: 10.1002/poi3.97
[4]	R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. https://doi.org/10.1017/S0950268800059896
[5]	L. M. Bettencourt, A. Cintrón-Arias, D. I. Kaiser, C. Castillo-Chávez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models, Phys. A, 364 (2006), 513–536. https://doi.org/10.1016/j.physa.2005.08.083 doi: 10.1016/j.physa.2005.08.083
[6]	M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167–256. https://doi.org/10.1137/S003614450342480 doi: 10.1137/S003614450342480
[7]	R. Chen, L. Kong, M. Wang, Modeling the dynamics of adoption and abandonment of multiple products, Math. Comput. Simul., 241 (2026), 868–889. https://doi.org/10.1016/j.matcom.2025.10.026 doi: 10.1016/j.matcom.2025.10.026
[8]	S. Wu, A. D. Sarma, A. Fabrikant, S. Lattanzi, A. Tomkins, Arrival and departure dynamics in social networks, in Proceedings of the Sixth ACM International Conference on Web Search and Data Mining, (2013), 233–242. https://doi.org/10.1145/2433396.2433425
[9]	B. Du, H. Wang, M. Liu, An information diffusion model in social networks with carrier compartment and delay, Nonlinear Anal. Model. Control, 23 (2018), 568–582. https://doi.org/10.15388/NA.2018.4.7 doi: 10.15388/NA.2018.4.7
[10]	T. Wang, J. He, X. Wang, An information spreading model based on online social networks, Phys. A, 490 (2018), 488–496. https://doi.org/10.1016/j.physa.2017.08.078 doi: 10.1016/j.physa.2017.08.078
[11]	R. Pastor-Satorras, A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200–3203. https://doi.org/10.1103/PhysRevLett.86.3200 doi: 10.1103/PhysRevLett.86.3200
[12]	N. Kimmel, L. Kong, M. Wang, Modeling the dynamics of user adoption and abandonment in online social networks, Math. Methods Appl. Sci., 48 (2025), 1853–1868. https://doi.org/10.1002/mma.10413 doi: 10.1002/mma.10413
[13]	C. Lei, Z. Lin, H. Wang, The free boundary problem describing information diffusion in online social networks, J. Differ. Equations, 254 (2013), 1326–1341. http://dx.doi.org/10.1016/j.jde.2012.10.021 doi: 10.1016/j.jde.2012.10.021
[14]	J. R. Graef, L. Kong, A. Ledoan, M. Wang, Stability analysis of a fractional online social network model, Math. Comput. Simul., 178 (2020), 625–645. https://doi.org/10.1016/j.matcom.2020.07.012 doi: 10.1016/j.matcom.2020.07.012
[15]	R. Chen, L. Kong, M. Wang, Stability analysis of an online social network model, Rocky Mt. J. Math., 53 (2023), 1019–1041. https://doi.org/10.1216/rmj.2023.53.1019 doi: 10.1216/rmj.2023.53.1019
[16]	Z. K. Zhang, C. Liu, X. X. Zhan, X. Lu, C. X. Zhang, Y. C. Zhang, Dynamics of information diffusion and its applications on complex networks, Phys. Rep., 651 (2016), 1–34. https://doi.org/10.1016/j.physrep.2016.07.002 doi: 10.1016/j.physrep.2016.07.002
[17]	G. Q. Sun, R. He, L. F. Hou, X. Luo, S. Gao, L. Chang, et al., Optimal control of spatial diseases spreading in networked reaction–diffusion systems, Phys. Rep., 1111 (2025), 1–64. https://doi.org/10.1016/j.physrep.2025.01.005 doi: 10.1016/j.physrep.2025.01.005
[18]	M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 61 (2015), 9–31.
[19]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)