Symmetry breaking by heating in a continuous opinion model

We study the critical behavior of a continuous opinion model, driven by kinetic exchanges in a fully connected population. Opinions range in the real interval $[-1,1]$, representing the different shades of opinions against and for an issue under debate. Individuals' opinions evolve through pairwise interactions, with couplings that are typically positive, but a fraction $p$ of negative ones is allowed. Moreover, a social temperature parameter $T$ controls the tendency of the individual responses toward neutrality. Depending on $p$ and $T$, different collective states emerge: symmetry broken (one side wins), symmetric (tie of opposite sides), and absorbing neutral (indecision wins). We find the critical points and exponents that characterize the phase transitions between them. The symmetry breaking transition belongs to the usual Ising mean-field universality class, but the absorbing-phase transitions, with $\beta =0.5$, are out of the paradigmatic directed percolation class. Moreover, ordered phases can emerge by increasing social temperature.

Figure 1

Steady value of the order parameter $\langle O\rangle$ versus $p$, for different values of $T$ indicated on the figure. (a) For high social temperature $T$, there is a transition from an asymmetric phase to an absorbing state where all opinions are neutral. (b) For low $T$, a transition from a symmetry-broken phase to a fluctuating symmetric one takes place. The population size is $N={10}^4$ and data are averaged over 100 simulations.

Figure 2

Steady value of the order parameter $\langle O\rangle$ versus the social temperature $T$, for different values of $p$ indicated on the figure. Averages are computed over 100 samples, and $N={10}^4$.

Figure 3

Critical points $p_c$ as a function of $T$, for the transitions between symmetry-broken, symmetric and neutral (absorbing) collective states. For more details about the distribution of opinions in each phase, see Fig. 6 in Appendix pp1. The arrows highlight the values for the discrete ($p_c^{II}=0.25$) and the continuous ($p_c^{II}\approx 0.34$) opinion models [7]. The full line is predicted by Eq. (10).

Figure 4

Steady value of the order parameter as a function of $\mathrm{\Delta }=|T-T_c|$ (a) for $p=0.1\left(T_c^I\approx 1.400\right)$ and $p=0.4\left(T_c^{III}\approx 1.155\right)$, and as a function of $\mathrm{\Delta }=|p-p_c|$ (b) for $T=1.3\left(p_c^I\approx 0.200\right)$. In all cases the order parameter goes to zero at the critical point as $\langle O\rangle \sim {\mathrm{\Delta }}^{\beta }$ with $\beta \approx 0.5$. The dashed lines with slope 0.5 are drawn for comparison. The solid lines are given by Eqs. (11) and (12). In all cases, system size is $N={10}^4$ and data were averaged over 100 realizations.

Figure 5

Absorbing-phase transition I. Scaling plots of the order parameter, based on the scaling relation Eq. (7). (a) $\mathrm{\Delta }=|T-T_c|$, for $p=0.1$ and different values of $T$ indicated on the figure, with data collapse obtained for $T_c^I\approx 1.400$, $\beta =0.5$, and $\nu =0.95$. (b) $\mathrm{\Delta }=|p-p_c|$, for $T=1.3$ and different values of $p$, with data collapse obtained for $p_c^I\approx 0.200$, $\beta =0.5$, and $\nu =0.98$. The insets show the original nonscaled data. Data are averages over 100 samples, for population size $N={10}^4$. The initial condition $o_i=1$ for all $i$ was used, although the final states are the same that for the random initial conditions used in other simulations.

Figure 6

Histograms of opinions at the steady state, for different values of $p$ and $T$. In all cases, for sufficiently high $T$, e.g., $T=1.43$, all opinions are neutral (hence, $\langle O\rangle =0$). As $T$ decreases, we observe the following behaviors: At $p=0.1$, when $T$ decreases below $T_c^I\left(0.1\right)\approx 1.400$, the distribution spreads asymmetrically and the peak shifts toward one of the extreme values. At $p=0.3$, transition I, from neutral to biased, is observed at $T_c^I\left(0.3\right)\approx 1.20$, but in this case, as $T$ decreases, the distribution becomes bimodal and tends to recover symmetry, condensing at the extreme values. At $p=0.4$, below $T_c^{III}\left(0.4\right)\approx 1.155$, the distributions spread, always symmetrically in this case, becoming bimodal at very low $T$ such that, also in this case, only extreme opinions in the same proportion survive in the low $T$ limit. The population size is $N={10}^4$ and data are averaged over 100 simulations.

Figure 7

Transition II. Finite-size scaling analysis of the quantities defined in Eqs. (3, 4, 5), based on the scaling relation Eq. (6), for $T=0.5$. The values of $N$ are indicated on the figure. Data collapse was obtained for $p_c^{II}\approx 0.321$, $\beta =0.5$, $\nu =2$, and $\gamma =\nu -2\beta =1.0$. The insets show the original nonscaled data.