Analyzing and forecasting the dynamics of Internet resource user sentiments based on the Fokker–Planck equation

Objectives. The study aims to theoretically derive the power law observed in practice for the distribution of characteristics of sociodynamic processes from the stationary Fokker–Planck equation and apply the non-stationary Fokker–Planck equation to describe the dynamics of processes in social systems.

Methods. During the research, stochastic modeling methods were used along with methods and models derived from graph theory, as well as tools and technologies of object-oriented programming for the development of systems for collecting data from mass media sources, and simulation modeling approaches.

Results. The current state of the comment network graph can be described using a vector whose elements are the average value of the mediation coefficient, the average value of the clustering coefficient, and the proportion of users in a corresponding state. The critical state of the network can be specified by the base vector. The time dependence of the distance between the base vector and the current state vector forms a time series whose values can be considered as the “wandering point” whose movement dynamics is described by the non-stationary Fokker–Planck equation. The current state of the comment graph can be determined using text analysis methods.

Conclusions. The power law observed in practice for the dependence of the stationary probability density of news distribution by the number of comments can be obtained from solving the stationary Fokker–Planck equation, while the non-stationary equation can be used to describe processes in complex network structures. The vector representation can be used to describe the comment network states of news media users. Achieving or implementing desired or not desired states of the whole social network can be specified on the basis of base vectors. By solving the non-stationary Fokker–Planck equation, an equation is obtained for the probability density of transitions between system states per unit time, which agree well with the observed data. Analysis of the resulting model using the characteristics of the real time series to change the graph of comments of users of the RIA Novosti portal and the structural parameters of the graph demonstrates its adequacy.

1. Du B., Lian X., Cheng X. Partial differential equation modeling with Dirichlet boundary conditions on social networks. Bound. Value Probl. 2018;2018(1):50. https://doi.org/10.1186/s13661-018-0964-4

2. Lux T. Inference for systems of stochastic differential equations from discretely sampled data: a numerical maximum likelihood approach. Ann. Finance. 2013;9(2):217–248. http://doi.org/10.1007/s10436-012-0219-9

3. Hurn A., Jeisman J., Lindsay K. Teaching an Old Dog New Tricks: Improved Estimation of the Parameters of Stochastic Differential Equations by Numerical Solution of the Fokker–Planck Equation. In: Dungey M., Bardsley P. (Eds.). Proceedings of the Australasian Meeting of the Econometric Society. 2006. The Australian National University, Australia. P. 1–36.

4. Elliott R.J., Siu T.K., Chan L. A PDE approach for risk measures for derivatives with regime switching. Ann. Finance. 2007;4(1):55–74. http://dx.doi.org/10.1007/s10436-006-0068-5

5. Orlov Y.N., Fedorov S.L. Nonstationary time series trajectories generation on the basis of the Fokker–Planck equation. TRUDY MFTI = Proceedings of MIPT. 2016;8(2):126–133 (in Russ.).]

6. Chen Y., Cosimano T.F., Himonas A.A., Kelly P. An Analytic Approach for Stochastic Differential Utility for Endowment and Production Economies. Comput. Econ. 2013;44(4):397–443. http://doi.org/10.1007/s10614-013-9397-4

7. Savku E., Weber G.-W. Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market. Ann. Oper. Res. 2020;132(6):1171–1196. https://doi.org/10.1007/s10479-020-03768-5

8. Krasnikov К.Е. Mathematical modeling of some social processes using game-theoretic approaches and making managerial decisions based on them. Russ. Technol. J. 2021;9(5):67–83 (in Russ.). https://doi.org/10.32362/2500-316X-2021-9-5-67-83 ]

9. Kirn S.L., Hinders M.K. Dynamic wavelet fingerprint for differentiation of tweet storm types. Soc. Netw. Anal. Min. 2020;10(1):4. https://doi.org/10.1007/s13278-019-0617-3

10. Hoffmann T., Peel L., Lambiotte R., Jones N.S. Community detection in networks without observing edges. Sci. Adv. 2020;6(4):1478. https://doi.org/10.1126/sciadv.aav1478

11. Pulipati S., Somula R., Parvathala B.R. Nature inspired link prediction and community detection algorithms for social networks: a survey. Int. J. Syst. Assur. Eng. Manag. 2021. https://doi.org/10.1007/s13198-021-01125-8

12. Dorogovtsev S.N., Mendes J.F.F. Evolution of networks. Adv. Phys. 2002;51(4):1079–1187. http://doi.org/10.1080/00018730110112519

13. Newman M.E.J. The structure and function of complex networks. SIAM Rev. 2003;45(2):167–256. https://doi.org/10.1137/S003614450342480

14. Dorogovtsev S.N., Mendes J.F.F., Samukhin A.N. Generic scale of the scale-free growing networks. Phys. Rev. E. 2001;63(6):062101. https://doi.org/10.1103/PhysRevE.63.062101

15. Golder S., Wilkinson D., Huberman B. Rhythms of Social Interaction: Messaging Within a Massive Online Network. In: Steinfield C., Pentland B.T., Ackerman M., Contractor N. (Eds.). Communities and Technologies. 2007. https://doi.org/10.1007/978-1-84628-905-7_3

16. Kumar R., Novak J., Tomkins A. Structure and evolution of online social networks. In: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and data Mining, KDD ’06. 2006. Р. 611–617. https://doi.org/10.1145/1150402.1150476

17. Mislove A., Marcon M., Gummadi K.P., Druschel P., Bhattacharjee B. Measurement and analysis of online social networks. In: Proceedings of the 7th ACM SIGCOMM Conference on Internet Measurement, IMC’07. 2007. Р. 29–42. https://doi.org/10.1145/1298306.1298311

18. Zhukov D., Khvatova T., Zaltsman A. Stochastic Dynamics of Influence Expansion in Social Networks and Managing Users’ Transitions from One State to Another. In: Proceedings of the 11th European Conference on Information Systems Management (ECISM 2017). 2017. Р. 322–329.

19. Zhukov D., Khvatova T., Millar C., Zaltcman A. Modelling the stochastic dynamics of transitions between states in social systems incorporating self-organization and memory. Technol. Forecast. Soc. Change. 2020;158:120134. https://doi.org/10.1016/j.techfore.2020.120134

20. Zhukov D., Khvatova T., Istratov L. A stochastic dynamics model for shaping stock indexes using self-organization processes, memory and oscillations. In: Proceedings of the European Conference on the Impact of Artificial Intelligence and Robotics (ECIAIR 2019). 2019. Р. 390–401.

21. Zhukov D.O., Zaltcman A.D., Khvatova T.Yu. Forecasting Changes in States in Social Networks and Sentiment Security Using the Principles of Percolation Theory and Stochastic Dynamics. In: Proceedings of the 2019 IEEE International Conference “Quality Management, Transport and Information Security, Information Technologies” (IT&QM&IS). 2019. Article number 8928295. Р. 149–153. https://doi.org/10.1109/ITQMIS.2019.8928295