# Modeling Sociodynamic Processes Based on the Use of the Differential Diffusion Equation with Fractional Derivatives

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the Problems of Modeling Non-Stationary Time Series and Modern Research in Dynamics of Social Processes

## 3. Selection of Data Sources and Collection of Statistics about User Activity on News Resources

## 4. Data Processing and Analysis of Observed Time Series—Objective Setting

^{H}, where C stands for a certain constant, τ stands for the number of observations (series levels) that make up the time series (TS) in question, and H is the Hurst exponent. The presence of fractures in the R/S (τ) dependence suggests that there are characteristic time scales and frequencies. The value H makes it possible to classify the time series by their behavior.

^{2}= 0.99.

^{2}= 0.89. In this case, H = 0.18 and, therefore, the observed time series is also non-persistent (ergodic).

- First, we can set the size of the sliding window (i.e., the time interval between the observed values of user activity on online resources), e.g., one day, two days, three days, etc., and from the time series select activity-related data for a given time range (a sliding window size).
- Furthermore, using the selected data, we can calculate amplitudes of changes in user activity on the online news resources for various selected time intervals (sliding window sizes). These data are presented in Figure 3.
- We should then sort the sets of values obtained for each of the measured intervals in ascending order (i.e., from negative to positive) and plot amplitude histograms for online news resources user activity deviations for each sliding window size.
- Based on the histograms, we can calculate distribution moments (mathematical expectation, dispersion, asymmetry, and excess) for the intervals selected for calculating amplitudes of online resources user activity deviations (sliding window sizes). Mathematical expectation, dispersion, asymmetry and excess can be calculated using the following formulas:

- 5.

^{2}(t) was calculated as an average value of the square-amplitude. With the normal distribution law, there would be a linear dependence on the sliding window size for the amplitude dispersion, as follows:

- The time series describing the discussed processes are non-stationary (the mathematical expectation and dispersion of the user news commenting activity change amplitude, in a complicated way, depends on the amplitude calculation interval (sliding window)).
- The analysis of the observed non-stationary time series shows that the processes and systems they describe have a short-term memory (the Hurst exponent is significantly less than 0.5).
- In the distribution of the user news commenting activity change amplitudes, asymmetry has a little value and the amplitude distribution is almost symmetrical.
- In the distribution of the user news commenting activity change amplitudes, there is a heavy tail: the probability plots will lie above the normal probability plot, and, therefore, these processes cannot be described by this law.
- When modeling dynamics of the processes in question and forecasting how the relevant time series may evolve, it is necessary to search for other approaches and models that would fully take into account their described properties.

## 5. Model of Sociodynamic Processes Based on the Use of Differential Diffusion Equation with Fractional Derivatives

#### 5.1. The Model’s Main Equation

#### 5.2. Model Analysis and Comparison with the Observed Data

**α**q order, by the value of amplitude x for the initial distribution, and does not depend on time t.

- for ”RIA Novosti”, y = −696x − 1.60, with the correlation coefficient R
^{2}= 0.97. - for ”Echo of Moscow”, y = −813x − 1.67, with the correlation coefficient R
^{2}= 0.99.

- for ”RIA Novosti”, y = −553x − 2.08, with the correlation coefficient R
^{2}= 0.93. - for ”Echo of Moscow”, y = −704x − 1.86, with the correlation coefficient R
^{2}= 0.99.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Nassirtoussi, A.K.; Wah, T.Y.; Ling, D.N.C. A novel FOREX prediction methodology based on fundamental data. Afr. J. Bus. Manag.
**2011**, 5, 8322–8330. [Google Scholar] - Anastasakis, L.; Mort, N. Exchange rate forecasting using a combined parametric and nonparametric self—Organising modelling approach. Expert Syst. Appl.
**2009**, 36, 12001–12011. [Google Scholar] [CrossRef] - Vanstone, B.; Finnie, G. Enhancing stockmarket trading performance with ANNs. Expert Syst. Appl.
**2010**, 37, 6602–6610. [Google Scholar] [CrossRef] - Vanstone, B.; Finnie, G. An empirical methodology for developing stockmarket trading systems using artificial neural networks. Expert Syst. Appl.
**2009**, 36, 6668–6680. [Google Scholar] [CrossRef] - Sermpinis, G.; Laws, J.; Karathanasopoulos, A.; Dunis, C.L. Forecasting and trading the EUR/USD exchange rate with gene expression and psi sigma neural networks. Expert Syst. Appl.
**2012**, 39, 8865–8877. [Google Scholar] [CrossRef] - Huang, S.-C.; Chuang, P.-J.; Wu, C.-F.; Lai, H.-J. Chaos-based support vector regressions for exchange rate forecasting. Expert Syst. Appl.
**2010**, 37, 8590–8598. [Google Scholar] [CrossRef] - Premanode, B.; Toumazou, C. Improving prediction of exchange rates using differential EMD. Expert Syst. Appl.
**2013**, 40, 377–384. [Google Scholar] [CrossRef] - Mabu, S.; Hirasawa, K.; Obayashi, M.; Kuremoto, T. Enhanced decision making mechanism of rule-based genetic network programming for creating stock trading signals. Expert Syst. Appl.
**2013**, 40, 6311–6320. [Google Scholar] [CrossRef] - Bahrepour, M.; Akbarzadeh, T.M.-R.; Yaghoobi, M.; Naghibi, S.M.-B. An adaptive ordered fuzzy time series with application to FOREX. Expert Syst. Appl.
**2011**, 38, 475–485. [Google Scholar] [CrossRef] - Preethi, P.G.; Uma, V.; Kumar, A. Temporal Sentiment Analysis and Causal Rules Extraction from Tweets for Event Prediction. Procedia Comput. Sci.
**2015**, 48, 84–89. [Google Scholar] [CrossRef] - Ren, Z.; Zeng, A.; Zhang, Y. Structure-oriented prediction in complex networks. Phys. Rep.
**2018**, 750, 1–5. [Google Scholar] [CrossRef] [Green Version] - Zhukov, D.; Khvatova, T.; Aleshkin, A.; Schiavone, F. Forecasting news events based on the model accounting for self-organisation and memory. In Proceedings of the 2021 IEEE Technology and Engineering Management Conference-Europe, TEMSCON-EUR, Virtual, 17–20 May 2021. Article number 94886342021. [Google Scholar]
- Zhukov, D.; Andrianova, E.; Trifonova, O. Stochastic diffusion model for analysis of dynamics and forecasting events in news feeds. Symmetry
**2021**, 13, 257. [Google Scholar] [CrossRef] - Fuentes, M.A. Non-Linear Diffusion and Power Law Properties of Heterogeneous Systems: Application to Financial Time Series. Entropy
**2018**, 20, 649. [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. (In Russian) [Google Scholar] [CrossRef] - Hurst, H.E. Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng.
**1951**, 116, 770–779. [Google Scholar] [CrossRef] - Mandelbrot, B.B. The Fractal Geometry of Nature; W.H. Freeman: New York, NY, USA, 1982. [Google Scholar]
- Mainardi, F. Waves and Stability in Continuous Media; Rionero, S., Ruggeri, T., Eds.; World Scientific: Singapore, 1994. [Google Scholar]
- Wyss, W.J. The fractional diffusion equation. J. Math. Phys.
**1986**, 27, 2782. [Google Scholar] [CrossRef] - Schneider, W.R.; Wyss, W.J. Fractional diffusion and wave equations. J. Math. Phys.
**1989**, 30, 134. [Google Scholar] [CrossRef] - Ilic, M.; Liu, F.; Turner, I.; Anh, V. Numerical approximation of a fractional-inspace diffusion equation, I. Fract. Calc. Appl. Anal.
**2005**, 8, 323–341. [Google Scholar] - Oldham, K.B.; Spanier, J. The Fractional Calculus; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- Xiao, J.; Guo, X.; Li, Y.; Wen, S.; Shi, K.; Tang, Y. Extended analysis on the global Mittag-Leffler synchronization problem for fractional-order octonion-valued BAM neural networks. Neural Netw.
**2022**, 154, 491–507. [Google Scholar] [CrossRef] - Xiao, J.; Zhong, S.; Wen, S. Unified Analysis on the Global Dissipativity and Stability of Fractional-Order Multidimension-Valued Memristive Neural Networks With Time Delay. IEEE Trans Neural Netw Learn Syst.
**2022**, 33, 5656–5665. [Google Scholar] [CrossRef] - Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. Inc.: New York, NY, USA, 1993. [Google Scholar]
- Samokhin, A.B. Methods and effective algorithms for solving multidimensional integral equations. Russ. Technol. J.
**2022**, 10, 70–77. [Google Scholar] [CrossRef]

**Figure 1.**Time Series of User Activity on the ”RIA Novosti” Website for Commenting on News from January 2013 until December 2020.

**Figure 2.**Hurst Exponent for Unscaled Time Series of User Activity on the ”RIA Novosti” Website (Line 1) and ”Echo of Moscow” (Line 2) for News Comments in the Observed Periods of Time.

**Figure 3.**Amplitude Histograms for ”RIA Novosti” (Line 1) and ”Echo of Moscow” (Line 2) User Activity Change When Commenting News for a Sliding Window of 1 Day (

**a**) and 20 Days (

**b**).

**Figure 4.**Dependence of Mathematical Expectation for ”RIA Novosti” (Line 1) and ”Echo of Moscow” (Line 2) User Activity Change Amplitudes When Commenting News on Amplitude Calculation Interval (Sliding Window).

**Figure 5.**Dependence of Dispersion for ”RIA Novosti” (Line 1) and ”Echo of Moscow” (Line 2) User News Commenting Activity Change Amplitudes on Amplitude Calculation Interval (Sliding Window).

**Figure 6.**Dependence of Probability Density $\rho \left(x,t\right)$ on the Observed Amplitude of Deviation x for the Modelled Example (curve 1—for conditional time of 75 units, curve 2—40 conditional units and curve 3—10 conditional units).

**Figure 7.**Amplitude Linearization for RIA Novosti (Line 1) and ”Echo of Moscow” (Line 2) User News Commenting Activity Change for a Sliding Window of 1 Day (

**a**) and 20 Days (

**b**).

**Figure 8.**Dependence of Theoretical Value of Mathematical Expectation for User News Commenting Activity Change Amplitudes on Amplitude Calculation Interval (Sliding Window) with Different Values of Fractional Derivatives α and β ((

**a**) for α = 0.92; β = 0.82 and (

**b**) for α = 1.10; β = 1.27).

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**MDPI and ACS Style**

Demidova, L.A.; Zhukov, D.O.; Andrianova, E.G.; Sigov, A.S.
Modeling Sociodynamic Processes Based on the Use of the Differential Diffusion Equation with Fractional Derivatives. *Information* **2023**, *14*, 121.
https://doi.org/10.3390/info14020121

**AMA Style**

Demidova LA, Zhukov DO, Andrianova EG, Sigov AS.
Modeling Sociodynamic Processes Based on the Use of the Differential Diffusion Equation with Fractional Derivatives. *Information*. 2023; 14(2):121.
https://doi.org/10.3390/info14020121

**Chicago/Turabian Style**

Demidova, Liliya A., Dmitry O. Zhukov, Elena G. Andrianova, and Alexander S. Sigov.
2023. "Modeling Sociodynamic Processes Based on the Use of the Differential Diffusion Equation with Fractional Derivatives" *Information* 14, no. 2: 121.
https://doi.org/10.3390/info14020121