Diffusion of binary opinions in a growing population with heterogeneous behaviour and external influence

We consider a growing population of individuals with binary opinions, namely, 0 or 1, that evolve in discrete time. The underlying interaction network is complete. At every time step, a fixed number of individuals are added to the population. The opinion of the new individuals may or may not depend on the current configuration of opinions in the population. Further, in each time step, a fixed number of individuals are chosen and they update their opinion in three possible ways: they organically switch their opinion with some probability and with some probability they adopt the majority or the minority opinion. We study the asymptotic behaviour of the fraction of individuals with either opinion and characterize conditions under which it converges to a deterministic limit. We analyze the behaviour of the limiting fraction as a function of the probability of new individuals having opinion 1 as well as with respect to the ratio of the number of people being added to the population and the number of people being chosen to update opinions. We also discuss the nature of fluctuations around the limiting fraction and study the transitions in scaling depending on the system parameters. Further, for this opinion dynamics model on a finite time horizon, we obtain optimal external influencing strategies in terms of when to influence to get the maximum expected fraction of individuals with opinion 1 at the end of the finite time horizon.