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 $p<p_c$, $T_c~\exp [\alpha (p)N]$, 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.

Figure 1

(a) The steady-state density $n_B$ of nodes in state $B$ as a function of committed fraction $p$ for complete graphs of different sizes, conditioned on survival of the system. Simulation results are from $100$ realizations of the binary agreement dynamics. (b) Movement of the stable fixed point (diamonds) and the saddle point (filled circles) in phase space as a function of committed fraction $p$ (see the text). The point at which they meet (coordinates shown) is indicated by the asterisk. The location of these points in phase space was obtained through fixed-point analysis of the mean-field equations (1) (see the Appendix). The inset shows the density of nodes in state $B$ at the stable (red diamonds) and unstable (blue filled circles) fixed points as $p$ is varied.

Figure 2

Trajectories [obtained from integration of the mean-field equations Eq. (1)] in the phase plane show the nature of flows from different regions of the phase plane into existing fixed points for (a) $p=0.05$ ($<p_c$) and (b) $p=0.1$ ($>p_c$).

Figure 3

(a) Mean consensus time $T_c$ for $p<p_c$ obtained by using the QS approximation. (b) Exponential scaling of $T_c$ with $N$, for $p<p_c$; mean consensus times (circles, squares) are obtained using the QS approximation. The lines are guides to the eye. (c) Logarithmic scaling of $T_c$ with $N$ for $p=0.3>p_c$; mean consensus times are obtained from simulations. The line shows the best linear fit to the data. (d) The rate $\alpha (p)$ of exponential growth of the consensus time with $N$ as a function of $p-p_c$ (see the text). Circles show the values of $\alpha (p)$ obtained for $p=0.2,0.3,0.4,0.5,0.6,0.7,0.8$ by considering the scaling of $T_c$ with $N$ for these values of $p$. The straight line shows a linear fit to the data plotted on a log-log scale.

Figure 4

The quasistationary distribution ${\mathrm{p̃}}_{\mathit{nm}}$ for $p=0.09$ and $N=100$.

Figure 5

(a) The steady-state density $n_B$ of nodes in state $B$ as a function of committed fraction $p$ for Erdős-Rényi graphs of different sizes with $\langle k\rangle =10$, conditioned on survival of the system. Symbols show mean values of $n_B$ obtained from $100$ simulations of different system sizes; the black line shows the mean density of nodes in state B obtained by integrating Eq. (1). (b) Scaling of $\Delta T_c(\langle k\rangle )$ (defined in the text) with $\langle k\rangle$; the line shows the best linear fit to the data. (c) The steady-state density $n_B$ of nodes in state $B$ as a function of committed fraction $p$ for Barabasi-Albert networks of different sizes with $\langle k\rangle =10$, conditioned on survival of the system [symbols as in (a)].