We investigate the emergence of extreme opinion trends in society by employing statistical physics modeling and analysis on polls that inquire about a wide range of issues such as religion, economics, politics, abortion, extramarital sex, books, movies and electoral vote. The surveys lay out a clear indicator of the rise of extreme views. The precursor is a nonlinear relation between the fraction of individuals holding a certain extreme view and the fraction of individuals that includes also moderates, e.g., in politics, those who are “very conservative” versus “moderate to very conservative” ones. We propose an activation model of opinion dynamics with interaction rules based on the existence of individual “stubbornness” that mimics empirical observations. According to our modeling, the onset of nonlinearity can be associated to an abrupt bootstrap-percolation transition with cascades of extreme views through society. Therefore, it represents an early-warning signal to forecast the transition from moderate to extreme views. Moreover, by means of a phase diagram we can classify societies according to the percolative regime they belong to, in terms of critical fractions of extremists and people’s ties.

A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann–Gibbs statistical mechanics [based on $S_1\mathrm{\equiv -k\int }\mathrm{d}\mathrm{u p(u)}\ln \mathrm{ p(u)].}$ Similarly, other classes of models point toward nonextensive statistical mechanics [based on $S_q\text{≡k[1−∫}\mathrm{d}{\mathrm{u[p(u)]}}^q\text{]/[q−1],}$ where the value of the entropic index $\mathrm{q\in }\mathfrak{R}$ depends on the specific model]. We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the type $\mathrm{u̇=f(u)+g(u)\xi (t)+\eta (t),}$ where $\mathrm{\xi (t)}$ and $\mathrm{\eta (t)}$ are independent zero-mean Gaussian white noises with respective amplitudes $M$ and $\mathrm{A.}$ This leads to the Fokker–Planck equation ${\partial }_t{\mathrm{P(u,t)=-\partial }}_u{\mathrm{[f(u)P(u,t)]+M\partial }}_u{\mathrm{\{g(u)\partial }}_u{\mathrm{[g(u)P(u,t)]\}+A\partial }}_{\mathrm{uu}}\mathrm{P(u,t).}$ Whenever the deterministic drift is proportional to the noise induced one, i.e., ${\mathrm{f(u)=-\tau g(u)g}}^{\prime }\mathrm{(u),}$ the stationary solution is shown to be ${\text{P(u,∞)∝{1−(1−q)β[g(u)]}}^2{\}}^{\text{1/(1−q)}}$ [with $\text{q≡(τ+3M)/(τ+M)}$ and $\text{β=(τ+M/2A)].}$ This distribution is precisely the one optimizing $S_q$ with the constraint ${\mathrm{〈[g(u)]}}^2{〉}_q\mathrm{\equiv \{\int }\mathrm{d}{\mathrm{u [g(u)]}}^2{\mathrm{[P(u)]}}^q\mathrm{\}/\{\int }\mathrm{d}{\mathrm{u [P(u)]}}^q\mathrm{\}=}\mathrm{const}.$ We also introduce and discuss various characterizations of the width of the distributions.