# Description of the Distribution Law and Non-Linear Dynamics of Growth of Comments Number in News and Blogs Based on the Fokker-Planck Equation

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## Abstract

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## 1. Introduction

## 2. Research Methods

## 3. A Brief Overview of Existing Studies of the Structure of Complex Social Systems and the Processes Observed

## 4. The Analysis of the Observed Statistics of Comments from Users of Newsfeed Resources and Blogs—Statement of the Research Problem

#### 4.1. Data Source Selection and Presentation

- News portal (The commercial radio station) is among the top 10 news sites in Russia and in July of 2021 took ninth place for attendance and seventh for user activity, also at the end of July 2021 ranking in the top eight of the cited radio stations and occupying first place through hyperlinks in social media at the end of August 2021 and fourth place according to the citation index in the media.
- The portal has various themes (presents news from the political, sporting, economic and scientific arenas, cultural orientation, etc.).
- The news portal has been in existence since 1990 and has established itself as a reliable, truthful and publicly available news source, and also publishes blogs of well-known media personalities.
- There is practically no pre-moderation of comments (pre-moderation applies only to new users or users who have previously violated the rules of the news portal), but there is post-moderation of discussions (the requirements for comments and prohibitions on their placement can be found at the link: https://echo.msk.ru/moderate.html (accessed on 13 October 2021)). Users can express different opinions (which do not have to coincide with the official position) and their comments are deleted only for violating the rules.

#### 4.2. Processing of Observed Data

- Gaussian distribution: $\rho \left(x\right)={e}^{-\frac{{x}^{2}}{2\xb7{\sigma}^{2}}}/\sigma \sqrt{2\pi}$
- Exponential distribution: $\rho \left(x\right)=a\xb7{e}^{-\alpha x}$
- Power distribution: $\rho \left(x\right)=\beta \xb7{x}^{-\gamma}$

- For the Gaussian distribution: $ln\left\{\rho \left(x\right)\right\}=-ln\left\{\sigma \sqrt{2\pi}\right\}-\frac{1}{2\xb7{\sigma}^{2}}\xb7{x}^{2}$
- For exponential distribution: $ln\left\{\rho \left(x\right)\right\}=ln\left\{a\right\}-\alpha x$
- For the power distribution: $ln\left\{\rho \left(x\right)\right\}=ln\left\{\beta \right\}-\gamma ln\left\{x\right\}$

_{i}/n = Σln{ρ(x

_{i})}/n). This is equal to 2.11 (0.13 << 2.11). Thus, the resulting regression is significant. The asymmetry characterizes the “skewness” of the distribution function, and for symmetric functions (for example, the normal distribution) it is zero (in our case, it is small and close to zero). The kurtosis characterizes the “tail” of the distribution. With large positive values for the kurtosis, the distribution function decreases more slowly with further distance from the average value than with small ones. If the excess value is greater than zero, the distribution density graph will lie above the normal distribution graph and, for less than zero, below the graph (in our case, this is small and very close to zero). Thus, from the data obtained, it can be concluded that the distribution of residuals is very close to normal, which confirms the conclusion that the natural logarithm of the proportion of commentators who wrote these comments linearly depends on the natural logarithm of the number of comments, which confirms the fulfillment of the power law.

## 5. Derivation of the Power Law of the Distribution of Comments from the Stationary Fokker-Planck Equation

## 6. A Model of the Nonlinear Dynamics of the Appearance of Comments Based on the Fokker-Planck Equation

_{,}the growth rate of the curve increases (see Figure 11).

## 7. Discussion

_{0}, D

_{0}and τ for various types of news, which can also allow prediction in the future what news may cause what user behavior, and how this may influence public opinion.

## 8. Conclusions

- The stationary distribution of news observed in practice by the number of comments to on it corresponds to the power law: $\rho \left(x\right)=\left[\gamma -1\right]{x}^{-\gamma}$, where $\rho \left(x\right)$ is the share of news items in their total number having $x$ comments, and $\gamma $ is the exponent.
- The dynamics of changes over time in the number of comments to a newsfeed or blog can have both an S-shaped form and a two-stage one, which may be due to a significant difference in the average time of appearance of comments at the second level (the time interval between the appearance of a comment at the first level and a comment on this comment), i.e., the value of the average delay.
- The power law of dependence observed in practice is the stationary probability density of the distribution of news by the number of comments (states $x$) which can be obtained from the solution of the stationary Fokker-Planck equation if some assumptions are made during its derivation. We assume that the coefficient $\mu \left(x\right)$ responsible in the Fokker-Planck equation for a purposeful change in the state of the system $x$ ($x$ is the current number of comments on the news) linearly depends on the state $x$, and the coefficient $D\left(x\right)$ responsible for a random change depends on $x$ quadratically. All this suggests that the Fokker-Planck equation can be used to describe processes in complex network structures.
- The solution of the unsteady Fokker-Planck equation under the assumptions of the linear dependence of $\mu \left(x\right)$ on the state of $x$ and the quadratic dependence of $D\left(x\right)$ on the state of $x$ allows us to obtain an equation for the probability density of transitions between the states of the system per unit of time, which are in good agreement with the observed data, taking into account the effect of the delay time between the appearance of the first level comment and the comment on this comment.
- The models developed based on the Fokker-Planck equation are in good agreement with the observed data, which makes it possible to create algorithms for monitoring and predicting the evolution of public opinion of users of news information resources.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Gardiner, C. Stochastic Methods: A Handbook for the Natural and Social Sciences; Springer: Berlin, Germany, 2009. [Google Scholar]
- Lux, T. Inference for systems of stochastic differential equations from discretely sampled data: A numerical maximum likelihood approach. Ann. Financ.
**2012**, 9, 217–248. [Google Scholar] [CrossRef] [Green Version] - 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 Financial Econometrics Handbook; Gregoriou, G., Pascalau, R., Eds.; Palgrave: London, UK, 2010. [Google Scholar]
- Elliott, R.J.; Siu, T.K.; Chan, L. A PDE approach for risk measures for derivatives with regime switching. Ann. Financ.
**2007**, 4, 55–74. [Google Scholar] [CrossRef] - Orlov, Y.N.; Fedorov, S.L. Generation of nonstationary time series trajectories based on the Fokker-Planck equation. WORKS MIPT
**2016**, 8, 126–133. [Google Scholar] - 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, 397–443. [Google Scholar] [CrossRef] - Savku, E.; Weber, G.-W. Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market. Ann. Oper. Res.
**2020**, 1–26. [Google Scholar] [CrossRef] - Andrianova, E.G.; Golovin, S.A.; Zykov, S.V.; Lesko, S.A.; Chukalina, E.R. Review of modern models and methods of analysis of time series of dynamics of processes in social, economic and socio-technical systems. Russ. Technol. J.
**2020**, 8, 7–45. [Google Scholar] [CrossRef] - Dorogovtsev, S.N.; Mendes, J.F.F. Evolution of networks. Adv. Phys.
**2002**, 51, 1079–1187. [Google Scholar] [CrossRef] [Green Version] - Newman, M.E.J. The structure and function of complex networks. SIAM Rev.
**2003**, 45, 167–256. [Google Scholar] [CrossRef] [Green Version] - Dorogovtsev, S.N.; Mendes, J.F.F.; Samukhin, A.N. Generic scale of the scale-free growing networks. Phys. Rev. E
**2001**, 63, 062101. [Google Scholar] [CrossRef] [Green Version] - Golder, S.A.; Wilkinson, D.M.; Huberman, B.A. Rhythms of social interaction: Messaging within a massive online network. In Communities and Technologies 2007; Steinfield, C., Pentland, B.T., Ackerman, M., Contractor, N., Eds.; Springer: London, UK, 2007; pp. 41–66. [Google Scholar]
- 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), Philadelphia, PA, USA, 20–23 August 2006; pp. 611–617. [Google Scholar]
- 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), San Diego, CA, USA, 24–26 October 2007; pp. 29–42. [Google Scholar]
- Pasa, L.; Navarin, N.; Sperdut, A. SOM-based aggregation for graph convolutional neural networks Neural Computing and Applications Neural Comput. Applic.
**2022**, 34, 5–24. [Google Scholar] - 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**. [Google Scholar] [CrossRef] - Airoldi, E.M.; Blei, D.M.; Fienberg, S.E.; Xing, E.P. Mixed membership stochastic blockmodels. J. Mach. Learn. Res.
**2008**, 9, 1981–2014. [Google Scholar] [PubMed] - Cho, Y.-S.; Steeg, G.V.; Galstyan, A. Co-evolution of selection and influence in social networks. In Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence (AAAI 2011), San Francisco, CA, USA, 7–11 August 2011. [Google Scholar]
- Sahafizadeh, E.; Ladani, B.T. The impact of group propagation on rumor spreading in mobile social networks. Phys. A Stat. Mech. Its Appl.
**2018**, 506, 412–423. [Google Scholar] [CrossRef] - Varma, V.S.; Morarescu, I.C.; Haye, Y. Analysis and control of multi-leveled opinions spreading in social networks. In Proceedings of the American Control Conference (ACC 2018), Milwaukee, WI, USA, 27–29 June 2018; pp. 3404–3409. [Google Scholar]
- López-Santamaría, L.-M.; Almanza-Ojeda, D.-L.; Gomez, J.C.; Ibarra-Manzano, M.-A. Age and Gender Identification in Unbalanced Social Media. In Proceedings of the 2019 International Conference on Electronics, Communications and Computers (CONIELECOMP), Cholula, Mexico, 27 February–1 March 2019. [Google Scholar] [CrossRef]
- Barberá, P. Less is More? How Demographic Sample Weights Can Improve Public Opinion Estimates Based on Twitter Data. 2016. Available online: http://pablobarbera.com/static/less-is-more.pdf (accessed on 21 December 2021).
- Luo, F.; Cao, G.; Mulligan, K.; Li, X. Explore Spatiotemporal and Demographic Characteristics of Human Mobility via Twitter: A Case Study of Chicago. Appl. Geogr.
**2016**, 70, 11–25. [Google Scholar] [CrossRef] [Green Version] - Sekara, V.; Stopczynski, A.; Lehmann, S. Fundamental structures of dynamic social networks. Proc. Natl. Acad. Sci. USA
**2016**, 113. [Google Scholar] [CrossRef] [Green Version] - Ubaldi, E.; Vezzani, A.; Karsai, M.; Perra, N.; Burioni, R. Burstiness and tie activation strategies in time-varying social networks. Sci. Rep.
**2017**, 7, srep46225. [Google Scholar] [CrossRef] - Yatim, A.F.M.; Wardhana, Y.; Kamal, A.; Soroinda, A.A.R.; Rachim, F.; Wonggo, M.I. A corpus-based lexicon building in Indonesian political context through Indonesian online news media. In Proceedings of the 2016 International Conference on Advanced Computer Science and Information Systems (ICACSIS), Malang, Indonesia, 15–16 October 2016. [Google Scholar] [CrossRef]
- Kirn, S.L.; Hinders, M.K. Dynamic wavelet fingerprint for differentiation of tweet storm types. Soc. Netw. Anal. Min.
**2020**, 10, 4. [Google Scholar] [CrossRef] - Karami, A.; Elkouri, A. Political Popularity Analysis in Social Media; Springer: Berlin, Germany, 2019; pp. 456–465. [Google Scholar]
- Koti, P.; Pothula, S.; Dhavachelvan, P. Age Forecasting Analysis—Over Microblogs. In Proceedings of the 2017 Second International Conference on Recent Trends and Challenges in Computational Models (ICRTCCM), Tindivanam, India, 3–4 February 2017; pp. 83–86. [Google Scholar] [CrossRef]
- Mukhamediev, R.I.; Yakunin, K.; Mussabayev, R.; Buldybayev, T.; Kuchin, Y.; Murzakhmetov, S.; Yelis, M. Classification of Negative Information on Socially Significant Topics in Mass Media. Symmetry
**2020**, 12, 1945. [Google Scholar] [CrossRef] - Ko, H.; Jong, Y.; Sangheon, K.; Libor, M. Human-machine interaction: A case study on fake news detection using a backtracking based on a cognitive system. Cogn. Syst. Res.
**2019**, 55, 77–81. [Google Scholar] - Bushman, B.; Whitaker, J. Media Influence on Behavior. Reference Module in: Neuroscience and Biobehavioral Psychology. 2017. Available online: http://scitechconnect.elsevier.com/neurorefmod/ (accessed on 24 November 2020).
- Bandari, R.; Asur, S.; Huberman, B.A. The Pulse of News in Social Media: Forecasting Popularity. arXiv
**2012**, arXiv:1202.0332v1. Available online: https://arxiv.org/pdf/1202.0332.pdf (accessed on 21 December 2021). - Willaert, T.; Van Eecke, P.; Beuls, K.; Steels, L. Building Social Media Observatories for Monitoring Online Opinion Dynamics. Soc. Media Soc.
**2020**, 6. [Google Scholar] [CrossRef] - Tran, C.; Shin, W.-Y.; Spitz, A. Community Detection in Partially Observable Social Networks. ACM Trans. Knowl. Discov. Data
**2021**, 16, 1–24. [Google Scholar] [CrossRef] - Chen, Z.; Li, X.; Bruna, J. Supervised community detection with line graph neural networks. In Proceedings of the 7th International Conference on Learning Representations (ICLR 2019), New Orleans, LA, USA, 6–9 May 2019. [Google Scholar]
- Hoffmann, T.; Peel, L.; Lambiotte, R.; Jones, N.S. Community detection in networks without observing edges. Sci. Adv.
**2020**, 6, eaav1478. [Google Scholar] [CrossRef] [Green Version] - Du, B.; Lian, X.; Cheng, X. Partial differential equation modeling with Dirichlet boundary conditions on social networks. Bound. Value Probl.
**2018**, 2018, 50. [Google Scholar] [CrossRef] - Liu, X.; He, D.; Liu, C. Modeling information dissemination and evolution in time-varying online social network based on thermal diffusion motion. Phys. A Stat. Mech. its Appl.
**2018**, 510, 456–476. [Google Scholar] [CrossRef] - Bomba, A.; Kunanets, N.; Pasichnyk, V.; Turbal, Y. Mathematical and computer models of message distribution in social networks based on the space modification of Fermi-Pasta-Ulam approach. Adv. Intell. Syst. Comput.
**2019**, 836, 257–266. [Google Scholar] - 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), Genoa, Italy, 14–15 September 2017; pp. 322–329. [Google Scholar]
- Sigov, A.S.; Zhukov, D.O.; Khvatova, T.Y.; Andrianova, E.G. A Model of Forecasting of Information Events on the Basis of the Solution of a Boundary Value Problem for Systems with Memory and Self-Organization. J. Commun. Technol. Electron.
**2018**, 63, 1478–1485. [Google Scholar] [CrossRef] - 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. Chang.
**2020**, 158, 120134. [Google Scholar] [CrossRef] - 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), Oxford, UK, 31 October–1 November 2019; pp. 390–401. [Google Scholar]
- Zhukov, D.; Khvatova, T.; Istratov, L. Analysis of non-stationary time series based on modelling stochastic dynamics considering self-organization, memory and oscillations. In Proceedings of the International Conference on Time Series and Forecasting (ITISE 2019), Granada, Spain, 25–27 September 2019; Volume 1, pp. 244–254. [Google Scholar]
- Khvatova, T.; Zaltsman, A.; Zhukov, D. Information processes in social networks: Percolation and stochastic dynamics. CEUR Workshop. In Proceedings of the 2nd International Scientific Conference “Convergent Cognitive Information Technologies”; Springer: Berlin/Heidelberg, Germany, 2017; Volume 2064, pp. 277–288. [Google Scholar]
- Zhukov, D.O.; Lesko, S.A. Stochastic self-organissation of poorly structured data and memory realisation in an information domain when designing news events forecasting models. In Proceedings of the 2nd IEEE International Conference on Big Data Intelligence and Computing, Auckland, New Zealand, 8–12 August 2016. [Google Scholar] [CrossRef]
- Zhukov, D.O.; Zaltcman, A.G.; Khvatova, T.Y. 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 and QM and IS), Sochy, Russia, 23–27 September 2019; pp. 149–153. [Google Scholar]

**Figure 1.**Density of distribution of commentators by their number of comments for the period from 1 January to 31 December 2020.

**Figure 5.**Linearization of the observed data for power distribution after cleaning unscrupulous users.

**Figure 6.**The observed dynamics of change over time, the number of comments on a news item of public interest that appeared on the portal https://echo.msk.ru/news/2626290-echo.html on 16 April 2020.

**Figure 7.**The observed dynamics of change over time, the number of comments on a news item of public interest that appeared on the portal https://echo.msk.ru/news/2740844-echo.html on 12 November 2020.

**Figure 8.**Dynamics of changes over time, the number of comments on a news item of public interest that appeared on the portal https://echo.msk.ru/news/2740844-echo.html on 12 November 2020, after removing the time gaps.

**Figure 9.**The observed dynamics of change over time, the number of comments on a news item of public interest that appeared on the portal https://echo.msk.ru/news/2571431-echo.html on 15 January 2020.

**Figure 10.**Dynamics of changes over time, the number of comments on a news item of public interest that appeared on the portal https://echo.msk.ru/news/2571431-echo.html on 15 January 2020, after removing the time gaps.

**Figure 11.**Dynamics of changes over time in the number of comments to the news in a simulation model based on the Fokker-Planck equation.

**Figure 12.**Dynamics of changes over time in the number of comments on the news in the simulation model based on the Fokker-Planck equation, considering two parallel processes.

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**MDPI and ACS Style**

Zhukov, D.; Perova, J.; Kalinin, V.
Description of the Distribution Law and Non-Linear Dynamics of Growth of Comments Number in News and Blogs Based on the Fokker-Planck Equation. *Mathematics* **2022**, *10*, 989.
https://doi.org/10.3390/math10060989

**AMA Style**

Zhukov D, Perova J, Kalinin V.
Description of the Distribution Law and Non-Linear Dynamics of Growth of Comments Number in News and Blogs Based on the Fokker-Planck Equation. *Mathematics*. 2022; 10(6):989.
https://doi.org/10.3390/math10060989

**Chicago/Turabian Style**

Zhukov, Dmitry, Julia Perova, and Vladimir Kalinin.
2022. "Description of the Distribution Law and Non-Linear Dynamics of Growth of Comments Number in News and Blogs Based on the Fokker-Planck Equation" *Mathematics* 10, no. 6: 989.
https://doi.org/10.3390/math10060989