Control over Recommendation Algorithms in Heterogeneous Modular Systems with Dynamic Opinions

The paper suggests a model-dependent theoretical framework for designing optimal ranking algorithms to achieve desirable macroscopic opinion configurations. We consider an opinion formation process in which agents communicate through stochastic pairwise interactions, with the outcomes of these interactions being a function of the interacting agents' opinions and individual attributes (types). For the model, we write a mean-field approximation (MFA)-a coarse-grained nonlinear ordinary differential equation-which accommodates network modularity and assortativity, agents' activity heterogeneity, and the curation of a ranking system that can prohibit interactions with opinion- and type-dependent probabilities. Upon MFA, we formulate a control problem for dynamically adjusting the ranking algorithm's parameters. The existence of a solution is proved, and certain properties of optimal controllers are derived. For the case of a two-element opinion alphabet, we obtain a solution to the control problem using finite-difference schemes. This solution holds for any number of agent types and does not depend on external factors, such as the influence of social bots. Numerical tests corroborate our findings and also enable us to investigate the control problem for high-dimension opinion spaces, wherein we consider two primary scenarios: depolarization of an initially polarized society and nudging a social system towards a fixed endpoint of an opinion spectrum.