Local interaction scale controls the existence of a nontrivial optimal critical mass in opinion spreading

We study a model of opinion formation where the collective decision of a group is said to happen if the fraction of agents having the most common opinion exceeds a threshold value, a critical mass. We find that there exists a unique nontrivial critical mass giving the most efficient convergence to consensus. In addition, we observe that for small critical masses, the characteristic time scale for the relaxation to consensus splits into two. The shorter time scale corresponds to a direct relaxation and the longer one can be explained by the existence of intermediate metastable states similar to those found in [P. Chen and S. Redner, Phys. Rev. E 71, 036101 (2005)]. This longer time scale is dependent on the precise condition for consensus—with a modification of the condition it can go away.

Figure 1

(Color online) Illustration of three different sizes of local-influence groups on a square lattice: the agent located on the central site can join a local-influence group of sizes (a) 9, (b) 13, and (c) 25.

Figure 2

(Color online) Average exit probability $E_q$ as a function of the relaxation time $t$ (in units of Monte Carlo step per site) for local-influence group of sizes (a) $G=9$ and (b) $G=13$, and different critical masses $n_0$. Initially, a fraction $q=52\mathrm{\%}$ agents are assigned the opinion “$+1$.” Each Monte Carlo time step corresponds to a sweep of group opinion updating in terms of a random sequence. The data are averages from 5000 independent realizations.

Figure 3

(Color online) (a) Average exit probability as a function of the relaxation time, similar to Fig. 2, but for a local-influence group of size $G=25$ and different critical masses. (b) is similar to (a), but for a larger system, $N=90000$.

Figure 4

(Color online) The probability mass function of consensus times for the system with $G=25$ and critical masses $n_0=13$, 21, and 22. The data are plotted logarithmic binned in log-log scale to highlight the tail behavior. We note that the long tail for $n_0=13$ displays approximately exponential decay, similar to those that have been noted before in Ref. [9] where a simple majority rule model is considered. Error bars represent the standard error (and omitted if smaller than the symbol size). The inset shows the optimal size $n_0^{\star }$ of the local-influence group versus $G$. For $G=9$, 13, and 25, the values of $n_0^{\star }$ are 8, 11, and 21, respectively. Dashed and dotted lines are displayed for comparison, which correspond to $n_0^{\star }=G-1$ and $n_0^{\star }=\left(G+1\right)/2$, respectively.

Figure 5

Snapshots of opinion configuration of the agents at different relaxation times $t$ in a typical realization of our threshold majority rule dynamics. The white and black sites correspond, respectively, to the agents holding opinion $+1$ or $-1$. The size of local-influence group is $G=25$ and the critical mass is $n_0=22$.