Nonequilibrium phase transition in the coevolution of networks and opinions

Models of the convergence of opinion in social systems have been the subject of considerable recent attention in the physics literature. These models divide into two classes, those in which individuals form their beliefs based on the opinions of their neighbors in a social network of personal acquaintances, and those in which, conversely, network connections form between individuals of similar beliefs. While both of these processes can give rise to realistic levels of agreement between acquaintances, practical experience suggests that opinion formation in the real world is not a result of one process or the other, but a combination of the two. Here we present a simple model of this combination, with a single parameter controlling the balance of the two processes. We find that the model undergoes a continuous phase transition as this parameter is varied, from a regime in which opinions are arbitrarily diverse to one in which most individuals hold the same opinion.

Figure 1

(Color online) An illustration of the model, with vertex shapes representing different opinions. At each time step the system is updated according to the process illustrated in panel (a) with probability $\varphi$ or panel (b) with probability $1-\varphi$. In (a) a vertex $i$ is selected at random and one of its edges—in this case the edge $(i,j)$—is rewired to a new vertex $j^{\prime }$ holding the same opinion as $i$. In (b) vertex $i$ adopts the opinion of one of its neighbors $j$.

Figure 2

(Color online) Histograms of community sizes in the consensus state for values of $\varphi$ above, at, and below the critical point in panels (a), (b), and (c), respectively. Values of the other parameters are $N=3200$, $M=6400$ (giving $\overline{k}=4$), and $\gamma =10$. In panel (b) the distribution appears to follow a power law for part of its range, with exponent $3.5\pm 0.3$, as indicated by the solid line. Numerical data are averaged over ${10}^4$ realizations for each value of $\varphi$ and binned logarithmically.

Figure 3

(Color online) Finite size scaling for $\overline{k}=4$ and $\gamma =10$. (a) Crossing plot used to determine the critical value ${\varphi }_c$ and exponent $a$. We find ${\varphi }_c=0.458\pm 0.008$ and $a=0.61\pm 0.05$. The inset shows a blow up of the region around the critical point. (b) Scaling collapse used to determine the exponent $b$, which is found to take the value $b=0.7\pm 0.1$. The data are averaged over ${10}^4$ realizations for each value of $\varphi$. Error bars are shown where they are larger than the symbol size.

Figure 4

(Color online) Values of ${\varphi }_c$ as a function of $\gamma$ for various $\overline{k}$ obtained by finite size scaling analyses using system sizes $N=200$, 400, 800, and 1600, and ${10}^4$ realizations for each size and set of parameter values. Note that the horizontal axis is logarithmic.

Figure 5

(Color online) Scaling of the average time $\tau$ to reach consensus. (a) Convergence time as a function of system size for $\varphi =1$. The straight line is a fit to a logarithmic form and indicates that $\tau \sim \ln N$. (b) Coefficient of variation of the convergence time as a function of $\varphi$. The vertical gray line denotes our best estimate of the critical point ${\varphi }_c=0.458$. (c) Scaling plot used to determine the dynamical exponent $z$. The crossing point falls at $\varphi =0.45\pm 0.02$, consistent with our other estimates of ${\varphi }_c$. The dynamical exponent is found to take the value $z=0.61\pm 0.15$. Parameter values are $\overline{k}=4$ and $\gamma =10$ in all panels. All data points are averaged over ${10}^4$ realizations. Symbols are the same as in Fig. 3. For the sake of clarity, system sizes $N=400$ and $N=1600$ are omitted in (b).