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 pc \approx 10%, there is a dramatic decrease in the time, Tc, taken for the entire population to adopt the committed opinion. In particular, for complete graphs we show that when p < pc, Tc \sim \exp(\alpha(p)N), while for p > pc, Tc \sim \ln N. We conclude with simulation results for Erd\H{o}s-R\'enyi random graphs and scale-free networks which show qualitatively similar behavior.