  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 \approx 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 \sim \exp(\alpha(p)N), while for
p > p_c, T_c \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.
