Consensus formation times in anisotropic societies

We developed a statistical mechanics model to study the emergence of a consensus in societies of adapting, interacting agents constrained by a social rule $B$. In the mean-field approximation, we find that if the agents' interaction $H_0$ is weak, all agents adapt to the social rule $B$, with which they form a consensus; however, if the interaction is sufficiently strong, a consensus is built against the established status quo. We observed that, after a transient time ${\alpha }_t$, agents asymptotically approach complete consensus by following a path whereby they neglect their neighbors' opinions on socially neutral issues (i.e., issues for which the society as a whole has no opinion). ${\alpha }_t$ is found to be finite for most values of the interagent interaction $H_0$ and temperature $T$, with the exception of the values $H_0=1$, $T\to \infty$, and the region determined by the inequalities $\beta <2$ and $2\beta H_0<1+\beta -\sqrt{1+2\beta -{\beta }^2}$, for which consensus, with respect to $B$, is never reached.

Figure 1

Distribution of the components (A9), with their correspondent boundaries $b_0$ (A7) and $b_1$ (A8), in the plane $(\beta ,H_0)$.

Figure 2

Logarithm of the transient time $ln\left({\alpha }_t\right)$ as a function of $\beta$ and $H_0$.

Figure 3

Diagram of the system at ${\alpha }_t$. We labeled the regions where the proposed solution $r_t^2=w_t$ is stable by ${\mathcal{A}}_{u_0}$ (where ${\alpha }_t\gg {\alpha }_{0,1}$), ${\mathcal{A}}_0$ and ${\mathcal{A}}_1$ (where, in both cases, ${\alpha }_t={\alpha }_{0,1}$), by $I$ where the proposed solution is instable and by $\mathcal{B}$ the region where $r^2<w$ for all $\alpha$. We also indicated the sign of $R$ according to (16) and (25).

Figure 4

Phase diagram of the system in the limit of $\alpha \to \infty$. Transitions between any two phases are always of the first order.