Modeling and Structural Analysis of Social Networks

Modern social networks are characterized by a complex non-random topology; however, existing research provides only a fragmented description of their structural properties within the Russian segment, leaving a gap in the validation of classical graph models. The relevance is driven by the need to develop precise mathematical analysis methods for managing information flows and countering threats in the digital environment. The aim of the work is the modeling and structural analysis of the “VK” social network using graph theory to identify key metrics of clustering, the “small-world” effect, and scalability. The study is based on the analysis of an anonymized data sample from the “VK” social network comprising over 1 million users. An undirected graph of friendship ties was constructed. Algorithms were applied to calculate the clustering coefficient, node degree distribution, average shortest path length, and modularity using the Louvain method. It was established that the “VK” network exhibits properties of a scale-free network with a power-law degree distribution and the presence of hubs, a high clustering coefficient (≈0.52), and a short average path length (<5 steps), which corresponds to the Barabási–Albert model and the “small-world” effect. Stable thematic communities with high structural closure were identified. The obtained results correlate with the findings of Milgram, Watts-Strogatz, and Barabási-Albert, confirming the universality of these models for Russian platforms. Research prospects involve the study of dynamic information diffusion processes and the application of graph neural networks for link prediction.

1. Milgram S. The small world problem. Psychology Today. 1967;1(1):60-67. URL: https://snap.stanford.edu/class/cs224w-readings/milgram67smallworld.pdf

2. Watts D., Strogatz S. Collective dynamics of ‘small-world’ networks. Nature. 1998;393:440-442. URL: https://doi.org/10.1038/30918 2020;393:440-442

3. Barabási A.-L., Albert R. Emergence of scaling in random networks. Science. 1999.286(5439):509-12. URL: https://doi.org/10.1126/science.286.5439.509

4. Erdős P., Rényi A. On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences. 2021;5:17-61.

5. Galaganova S.G., Turusina T.V. Technologies for analyzing social networks in order to identify social trends. Human Сapital. 2023;1(169):121-136. (In Russ.). URL: https://doi.org/10.25629/HC.2023.01.14

6. Zhurina A.A., Kochkarov A.A., Kochkarov R.A. Analysis of structural characteristics of social networks. Social Networks and Internet Technologies. 2023;21(5):63-72. (In Russ.). URL: https://doi.org/10.18127/j20700814202305-08

7. Kochkarov R.A., Cherkasov V.V., Timoshenko A.V., Kochkarov A.A., Martynov N. S., Bodrov A. O. Structural graph visualization of social networks and community research in them. Social Networks and Internet Technologies. 2021;8(1):169-176. (In Russ.). URL: https://www.elibrary.ru/sgdrqn

8. Kochkarov A. A., Kalashnikov N. V., Kochkarov R. A. Comparative analysis of algorithms for identifying communities in complex network systems using the example of social networks. Software Products and Systems. 2020;(2):349-356. (In Russ.). URL: https://doi.org/10.15827/0236-235X.130.349-356

9. Badryzlov V.A., Sideltsev V.V. Evaluation of the effectiveness of information dissemination in social networks using simulation. Creative Economy. 2018;12(9):1359-1372. (In Russ.). URL: https://doi.org/10.18334/ce.12.9.39389

10. Ying R., He R., Chen K., Eksombatchai P., Hamilton W.L., Leskovec J. Graph Convolutional Neural Networks for Web-Scale Recommender Systems. In KDD ’18: The 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, August 19-23, 2018, London, United Kingdom. NY: ACM. URL: https://doi.org/10.1145/3219819.3219890

11. Kochkarov R.A., Kochkarov A.A., Sennikova L. I. Classification of multicriteria problems on multi-weighted prefractal graphs. In: Actual Problems of Applied Mathematics, Computer Science and Mechanics. Voronezh: VSU Publishing House; 2017:112-119. (In Russ.). URL: https://www.elibrary.ru/zgpfej