Phase change in an opinion-dynamics model with separation of time scales

We define an opinion formation model of agents in a one-dimensional ring, where the opinion of an agent evolves due to its interactions with close neighbors and due to its either positive or negative attitude toward the overall mood of all the other agents. While the dynamics of the agent’s opinion is described with an appropriate differential equation, from time to time pairs of agents are allowed to change their locations to improve the homogeneity of opinion (or comfort feeling) with respect to their short-range environment. In this way the timescale of transaction dynamics and that of environment update are well separated and controlled by a single parameter. By varying this parameter we discovered a phase change in the number of undecided individuals. This phenomenon arises from the fact that too frequent location exchanges among agents result in frustration in their opinion formation. Our mean field analysis supports this picture.

Figure 1

Diagram showing the exchange process used in the model (for $m=1$). The randomly chosen undecided nodes $i$ and $j$ have $q_{\mathit{ij}}^{(1)}<p_{\mathit{ij}}^{(1)}$; thus they are exchanged. Observe that the grayscale representing the opinion variable is more uniform after exchange.

Figure 2

(a) Average number of undecided agents as a function of the number of transactions per generation $g$. The purple squares are the numerical results after ${10}^7$ transactions, and the red circles are the agents who will never get a decision, according to a linear analysis. The corresponding mean field predictions are shown as continuous lines. (b) The average fraction of undecided agents with negative $\alpha$ (purple squares) and its corresponding mean field prediction (continuous line). (c) Average cluster size as a function of $g$. Observe the plateau around $g_c\approx 1.7\times {10}^3$.

Figure 3

Left panel: Time evolution of the absolute value of opinion averaged over 100 realizations of the ring of $N=5000$ agents for $g\to \infty$. The corresponding mean field prediction is shown as a continuous line. Right panels: Time history for a sample of 40 agents in a single realization for three different values of $g=7\times {10}^2,4\times {10}^3,4\times {10}^5$ (ordered from top to bottom), chosen to correspond to $g<g_c$, $g~g_c$, and $g>g_c$, around $g_c$ obtained from Fig. 2. The vertical dotted lines indicate the moments when there is a change of time regime in the dynamics, as explained in the text.

Figure 4

On the left-hand side we show a semi-log plot of the cluster size distribution for three values of $g=7\times {10}^2,4\times {10}^3,4\times {10}^5$ (green squares, purple circles, and light blue diamonds, respectively). The distribution for the initial random ring is shown for comparison as a dotted blue line. The panels on the right-hand side show the number distribution of $\alpha$ for 100 realizations of the same three values of $g$ (ordered from top to bottom) and for three different ranges of cluster size $s$: red circles for $s\in [1,5)$, green squares for $s\in [5,15)$, and blue diamonds for $s\in [15,5000]$.