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 \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.

💡 Research Summary

The paper investigates how a small, stubborn minority can overturn the prevailing majority opinion in a population. Using a binary agreement model—a variant of the Naming Game—agents can hold opinion A, opinion B, or both (state AB). At each discrete time step a random speaker and listener are selected; the speaker broadcasts one of its opinions, and the listener updates its state according to simple rules: if it already possesses the spoken opinion it discards the other, otherwise it adds the new opinion to its repertoire. A fraction p of agents are designated as “committed”: they permanently hold opinion A, never change, and can only influence others.

The authors first analyze the model on a complete graph (mean‑field limit). By writing rate equations for the densities n_A, n_B, and n_AB they identify fixed points and their stability as a function of p. For any p there is always the absorbing consensus fixed point (all uncommitted agents adopt A). However, when p is below a critical value p_c ≈ 0.10, two additional fixed points appear: a stable “active” fixed point where a non‑zero fraction of agents remain in state B, and an unstable saddle point. As p increases, these two collide at p_c in a classic first‑order phase transition, after which only the consensus fixed point remains. The order parameter chosen is n_B, which jumps discontinuously to zero at p_c.

In finite systems the active fixed point becomes metastable: the system lingers near it for a long time before a large fluctuation drives it to the absorbing state. To quantify the consensus time T_c the authors employ a quasi‑stationary (QS) approximation. Starting from the master equation for the joint probability P(n,m) (n agents in A, m in B), they condition on survival (i.e., not yet reached consensus) and iteratively solve for the QS distribution. The decay rate λ of the survival probability is obtained from the probability flux into the absorbing state, and the mean consensus time follows as T_c ≈ 1/λ. This analytical framework yields two distinct scaling regimes: for p < p_c, T_c grows exponentially with system size, T_c ∼ exp