Dynamical affinity in opinion dynamics modeling

We propose here a model to simulate the process of opinion formation, which accounts for the mutual affinity between interacting agents. Opinion and affinity evolve self-consistently, manifesting a highly nontrivial interplay. A continuous transition is found between single and multiple opinion states. Fractal dimension and signature of critical behavior are also reported. A rich phenomenology is presented and discussed with reference to corresponding psychological implications.

Figure 1

Left: Typical evolution of the opinion vs time, i.e., number of iterations. Right plot: Final affinity matrix. Here $\sigma =0.02$, $\Delta O_c=0.5$, and ${\alpha }_c=0.5$. Initial opinions are (random) uniformly distributed within the interval [0,1]. ${\alpha }_{ij}^0$ is initialized with uniform (random) values between 0 and 0.5. Here, ${\beta }_1={\beta }_2=1000$.

Figure 2

Average number of clusters as function of the rescaled quantity ${\left(\sigma {\alpha }_c\right)}^{-1/2}$. A phase transition is found at ${\left(\sigma {\alpha }_c\right)}^{-1/2}\simeq 20$. Above the transition, histograms of the number of clusters are computed and enclosed as insets in the main frame: Symbols refer to the numerics; solid lines are fitted interpolation. Here, $\Delta O_c=0.5$. The variables $O_i^0$ and ${\alpha }_{ij}^0$ are initialized as described in the caption of Fig. 1.

Figure 3

Main plot: Typical evolution in the monoclustered phase. Inset: $N_b$ vs $l=2^{-n_b}$ in log-log scale. For the choice of the parameters refer to the caption of Fig. 2

Figure 4

(Color online) Main frame: Histogram of $\delta O$, as follows from the numerics ($N=100$, averaged over 1000 independent realizations), plotted in log-log scale (symbols). The solid line is a guide for the eye. Inset: Cumulative distribution of the differences $\delta O$, in log-log scale.

Figure 5

(Color online) Rescaled convergence time $T_c^{\sigma }\left({\alpha }_c\right)/T_c^{\sigma }\left(0\right)$ is plotted as function of ${\alpha }_c$. Different symbols refer to different values of the social temperature $\sigma$; see legend. The dashed line stands for the theoretical prediction (7). The solid line is a numerical fit based on Eq. (7), where ${\alpha }_{\text{max}}$ is replaced by the effective value ${\alpha }_{\text{eff}}=0.66$ (see main text for further details).