Discrete opinion dynamics on networks based on social influence

A model of opinion dynamics based on social influence on networks was studied. The opinion of each agent can have integer values i = 1, 2, ..., I and opinion exchanges are restricted to connected agents. It was found that for any I >= 2 and self-confidence parameter 0 <= u < 1, when u is a degree-independent constant, the weighted proportion < q(i)> of the population that hold a given opinion i is a martingale, and the fraction q(i) of opinion i will gradually converge to < q(i)>. The tendency can slow down with the increase of degree assortativity of networks. When u is degree dependent, < q(i)> does not possess the martingale property, however q(i) still converges to it. In both cases for a finite network the states of all agents will finally reach consensus. Further if there exist stubborn persons in the population whose opinions do not change over time, it was found that for degree-independent constant u, both q(i) and < q(i)> will converge to fixed proportions which only depend on the distribution of initial obstinate persons, and naturally the final equilibrium state will be the coexistence of diverse opinions held by the stubborn people. The analytical results were verified by numerical simulations on Barabasi-Albert (BA) networks. The model highlights the influence of high-degree agents on the final consensus or coexistence state and captures some realistic features of the diffusion of opinions in social networks.