Identification of the Activity Function of Social Network Users in a Linear Diffusion Model

Introduction. Improving the accuracy of mathematical models for disseminating information in social networks is directly related to the ability to correctly identify their parameters. In numerous papers, the fundamental complexity of this problem is actually bypassed by substituting the direct identification of the desired functions for the selection of parameters for their heuristic approximations, which inevitably leads to a decrease in both the accuracy and universality of the model. In the linear diffusion model describing the spatiotemporal dynamics of information, one of the key parameters is the function characterizing user activity. The objective of this study includes the development and numerical implementation of an algorithm for direct parametric identification of user activity functions based on a direct extreme approach, which makes it possible to completely abandon heuristic approximations, and the evaluation of its computational efficiency in comparison to the classical gradient method.

Materials and Methods. A direct extreme approach was used to solve the parametric identification problem. Unlike the classical steepest descent technique, the proposed method with adjustable descent direction adapted the search trajectory to local features of the quality functional through introducing a control parameter. The numerical solution to the direct and adjoint problems was implemented using an implicit finite-difference scheme. The method was verified using synthetic data.

Results. For the identification algorithm, an analytical expression of the gradient of the target functional was obtained through the solution to the adjoint problem. The identifiability limits of the desired parameter conditioned by the inertia of the diffusion process and the network response time were determined. A comparative study of gradient algorithms was conducted. The classical steepest descent approach demonstrated slow and uneven convergence, requiring 13,217 iterations to reach the stopping criterion, whereas the method with adjustable descent direction provided convergence to the same level of accuracy in 376 iterations.

Discussion. The obtained results confirm the theoretical assumptions about the need to take into account the spatial heterogeneity of the functional gradient when solving infinite-dimensional optimization problems. The classical gradient technique exhibits low efficiency in reconstructing nonstationary parameters due to gradient nonuniformity, while the method with adjustable descent direction reaches uniform and rapid convergence. This demonstrates that adapting the algorithm to the specifics of an infinite-dimensional problem is a key success factor. The main contribution of the research is the development of a computing apparatus for the direct determination of functional parameters, which expands the methodological arsenal for analyzing systems described by partial differential equations.

Conclusion. The key findings of this research are the development and verification of an efficient algorithm for direct identifying user activity functions in a linear diffusion model of a social network. The practical significance consists in the creation of more accurate and interpretable tools for modeling information flows without resorting to a priori approximations. The developed algorithm has demonstrated significant advantages in speed and convergence. However, the interpretation of the physical meaning of the identified function within this model requires further development. A promising direction is the application of the method to more sophisticated models that take into account the spatial heterogeneity of user activity, as well as its extension to the identification of the function vector.

1. Mei Li, Xiang Wang, Kai Gao, Shanshan Zhang. A Survey on Information Diffusion in Online Social Networks: Models and Methods. Information. 2017;8(4):118. https://doi.org/10.3390/info8040118

2. Glukhov AI, Shishlenin MA, Trusov NV. Modeling the Dynamics of Social Protests: Mean Field Games and Inverse Problems. Differential Equations. 2025;61(6):802–822. https://doi.org/10.7868/S3034503025060067

3. Alshahrani M, Zhu Fuxi, Sameh A, Mekouar S, Sheng Huang. Efficient Algorithms Based on Centrality Measures for Identification of Top-K Influential Users in Social Networks. Information Sciences. 2020;527:88–107. https://doi.org/10.1016/j.ins.2020.03.060

4. Dritsas E, Trigka M. Machine Learning in Information and Communications Technology: A Survey. Information. 2025;16(1):8. https://doi.org/10.3390/info16010008

5. Haiyan Wang, Feng Wang, Kuai Xu. Modeling Information Diffusion in Online Social Networks with Partial Differential Equations. Cham: Springer; 2020. 144 p. https://doi.org/10.48550/arXiv.1310.0505

6. Ying Hu, Rachel Jeungeun Song, Min Chen. Modeling for Information Diffusion in Online Social Networks via Hydrodynamics. IEEE Access. 2017;5:128–135. https://doi.org/10.1109/ACCESS.2016.2605009

7. Feng Wang, Haiyan Wang, Kuai Xu, Jianhong Wu, Xiaohua Jia. Characterizing Information Diffusion in Online Social Networks with Linear Diffusive Model. In: Proc. IEEE 33rd International Conference on Distributed Computing Systems. New York City: IEEE; 2013. P. 307–316. https://doi.org/10.1109/ICDCS.2013.14

8. Zvonareva TA, Kabanikhin SI, Krivorotko OI. Numerical Algorithm for Source Determination in a Diffusion–Logistic Model from Integral Data Based on Tensor Optimization. Computational Mathematics and Mathematical Physics. 2023;63(9):1513–1523. https://doi.org/10.31857/S0044466923090193

9. Krivorotko O, Kabanikhin S, Shuhua Zhang, Kashtanova V. Global and Local Optimization in Identification of Parabolic Systems. Journal of Inverse and Ill-Posed Problems. 2020;28(6):899–913. https://doi.org/10.1515/jiip-2020-0083

10. Zoppoli R, Sanguineti M, Gnecco G, Parisini Th. The Basic Infinite-Dimensional or Functional Optimization Problem. In book: Neural Approximations for Optimal Control and Decision. Cham: Springer; 2020. P. 1–38. https://doi.org/10.1007/978-3-030-29693-3_1

11. Tolstykh VK. Direct Extreme Approach for Optimizing Distributed Parameter Systems. Donetsk: Yugo-Vostok; 1997. 177 p. (In Russ.)

12. Tolstykh MA. Identifying the Capacity of a Social Network. Moscow University Computational Mathematics and Cybernetics. 2024;48:59–64. https://doi.org/10.3103/S0278641924010084

13. Miele A. Theory of Optimum Aerodynamic Shapes. New York: Academic Press; 1965. 455 p.

14. Marchuk GI, Shutyaev VP. Adjoint Equations and Iterative Algorithms in Problems of Variational Data Assimilation. Proceedings of the Steklov Institute of Mathematics (Supplement Issues). 2012;276(2):138–152. https://doi.org/10.1134/S0081543812020113

15. Kabanikhin SI, Krivorotko OI. Optimization Methods for Solving Inverse Problems of Immunology and Epidemiology. Journal of Computational Mathematics and Mathematical Physics. 2020;60(4):590–600. https://doi.org/10.31857/S0044466920040109.