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 < pc, T(c) ~ exp [α(p)N], whereas for p>p(c), T(c) ~ ln N. We conclude with simulation results for Erdős-Rényi random graphs and scale-free networks which show qualitatively similar behavior.