Phase transition in a network model of social balance with Glauber dynamics

We study the evolution of a social network with friendly or enmity connections into a balanced state by introducing a dynamical model with an intrinsic randomness, similar to Glauber dynamics in statistical mechanics. We include the possibility of the tension promotion as well as the tension reduction in our model. Such a more realistic situation enables the system to escape from local minima in its energy landscape and thus to exit out of frozen imbalanced states, which are unwanted outcomes observed in previous models. On the other hand, in finite networks the dynamics takes the system into a balanced phase, if the randomness is lower than a critical value. For large networks, we also find a sharp phase transition at the initial positive link density of ${\rho }_0^{*}=1/2$, where the system transitions from a bipolar state into a paradise. This modifies the gradual phase transition at a nontrivial value of ${\rho }_0^{*}\simeq 0.65$, observed in recent studies.

Figure 1

Four distinct configurations of elementary units of the network. Solid lines show friendship links, and dashed lines represent hostility links. The triads with energy $-1$ ($+1$) are balanced (imbalanced).

Figure 2

The time evolution of (a) the total energy, $U$, and (b) the density of positive links, $\rho$, with different initial conditions for $\beta =0$. (c) The trajectory of this dynamics in $U-\rho$ space, for two initial conditions of ${\rho }_0=0$ and 1. The network size for all plots is $N=128$. Note that the dashed lines in (a), (b), and (c) show our analytical solutions of Eqs. (12), (15), and (16), respectively.

Figure 3

The time evolution of (a) the total energy, $U$, and (b) the density of positive links, $\rho$, with different initial conditions of ${\rho }_0\le 1/2$ for $\beta \to \infty$. The dashed lines show our analytical solutions, Eqs. (17) and (18), which are not in agreement with the simulations. The network size for all plots is $N=128$.

Figure 4

The time evolution of (a) the total energy, $U$, and (b) the density of positive links, $\rho$, with different initial conditions of ${\rho }_0>1/2$ for $\beta \to \infty$. The dashed lines show our analytical solutions, Eqs. (17) and (18), which are in complete agreement with the simulations. The network size for all plots is $N=128$.

Figure 5

The trajectory of the dynamics in $U-\rho$ space, for $\beta \to \infty$ with five different initial conditions. Note that representative points with $U=-1$ illustrate the balanced absorbing states. The dashed line represents our analytical solution of Eq. (16) for a fully uncorrelated trajectory. Note that as $\rho \to 1/2$, the deviation from the random trajectory increases. The network size is $N=128$.

Figure 6

(a) The $\beta$ dependency of large time behaviors of the total energy, $U$, for ${\rho }_0=0.6$. Different network sizes are shown with different symbols. For large networks, a nearly sharp transition occurs at some values of $\beta$, which goes to infinity as $N\to \infty$. (b) The order parameter, $\delta$, defined as the average link value, $〈s〉$, as a function of ${\rho }_0$, for different network sizes of $N=8$, 16, 32, 64, 128, 256, and 512, at $\beta =3$. There is a sharp transition at ${\rho }_0=1/2$ for large networks.