The article studies consensus formation problems in dynamic models of opinion dynamics in social networks with stubborn agents, which include external players influencing the agents’ opinions. The goal of each player is to maintain the agents’ opinion as close as possible to a given target value, where these values may differ for different players. The dynamics in agents’ opinions is described by a system of linear difference equations, and the players’ payoff functions have a quadratic form. The considered dynamic game belongs to the class of linear-quadratic games in discrete time. To find the optimal control at each stage, the Bellman equation is used, which allows one to determine the optimal trajectory of opinion dynamics and an analytical expression for the Nash equilibrium. When the degree of stubbornness of agents is unknown, the article proposes a new approach that uses two-level optimization and then the Euler equations to solve the inverse dynamic game problem based on observed data, which allows one to determine the levels of stubbornness of agents. Numerical experiments demonstrate the influence of the degree of stubbornness of agents and various structures of network interaction on the consensus formation process.
Keywords: opinion dynamics, social network, Bellman equation, inverse dynamic game
Received September 22, 2025
Revised December 01, 2025
Accepted December 15, 2025
Funding Agency: The research of the first and fourth authors was supported by the National Natural Science Foundation of China (No. 72171126) and Systems Science Plus Joint Research Program of Qingdao University (No. XT2024301).
Yanshan Liu, PhD student, Researcher, School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong, 266071 China, e-mail: 2021020269@qdu.edu.cn
Vladimir V. Mazalov, PhD, Prof., Saint Petersburg State University, Saint Petersburg, 198504 Russia; Institute of Applied Mathematical Research, Karelian Scientific Centre RAS, Petrozavodsk, 185910, Russia, e-mail: v.v.mazalov@spbu.ru
Hongwei Gao, PhD, Prof., School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong, 266071 China, e-mail: gaohongwei@qdu.edu.cn
Yajin Chen, PhD student, Researcher, School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong, 266071, China, e-mail: chenyajin@qdu.edu.cn
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Cite this article as: Yanshan Liu, Vladimir V. Mazalov, Hongwei Gao, Yajin Chen. Inverse Dynamic Game Model of Opinion Dynamics in a Social Network with Stubborn Agents. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 130–147.
