Social consensus through the influence of committed minorities

We show how the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed agents who consistently proselytize the opposing opinion and are immune to influence. Specifically, we show that when the committed fraction grows beyond a critical value p_c≈10%, there is a dramatic decrease in the time T_c taken for the entire population to adopt the committed opinion. In particular, for complete graphs we show that when p<p_c, T_c~exp[α(p)N], whereas for p>p_c, T_c~lnN. We conclude with simulation results for Erdős-Rényi random graphs and scale-free networks which show qualitatively similar behavior.