Opinion dynamics in a social network

Dating back to the 60's, and particularly since the advent and successful proliferation of social media, a significant effort has been devoted by the research community to the development of mathematical models describing the evolution of opinions/beliefs in an ecosystem composed of socially interacting individuals. The goal is to gain insights into collective dominant social behaviors and into the impact of different components of the system while being able to predict possible outcomes of social interactions. Recently, our Roberto Tempo published on Science a paper pointing out how political opinions can be steered by exploiting correlations between different subjects.
A typical way to model social interactions among individuals (hereinafter also called agents) is to use static or dynamic graphs, reflecting the social structure of the system. In such a representation, often agents directly interact only with their neighbors, varying their opinions for effect of pairwise ``attractive'' interactions. In a most typical model, the opinion of agent i after interaction with agent j is a linear combination of the current opinions of the two interacting agents, where the coefficients of the combinations depend on the degree of stubbornness of agent i. Often, each agent is also endowed with a prejudice, which does not get influenced by interactions and contributes to form her opinion.
In our work, we have adopted a fluid model in which, to keep into account a large number of agents, an opinion density has been defined, in the same way as, in statistical physics, probability theory is used as the mathematical tool to deal with large populations of particles. As a fundamental novelty, to model the underlying graph structure of the social network, we have introduced the personality, which is a variable on which the other parameters (degree of stubbornness, prejudice, strength of pairwise interactions) depend. With this model, the opinion density satisfies a partial differential equation similar to the Fokker-Planck equation.
If certain conditions are satisfied, we have been able to find the entire time evolution of the opinion density. In more general settings, we are able to predict the structure of the stationary solution, i.e., the solution after any transitory behavior has faded out. As a result, in most cases, we are able to predict whether the agents will reach a general consensus, or if, instead, some non-communicating clusters with different opinions will be formed.
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