Mean-field solution of structural balance dynamics in nonzero temperature

In signed networks with simultaneous friendly and hostile interactions, there is a general tendency to a global structural balance, based on the dynamical model of links status. Although the structural balance represents a state of the network with a lack of contentious situations, there are always tensions in real networks. To study such networks, we generalize the balance dynamics in nonzero temperatures. The presented model uses elements from Boltzmann-Gibbs statistical physics to assign an energy to each type of triad, and it introduces the temperature as a measure of tension tolerance of the network. Based on the mean-field solution of the model, we find out that the model undergoes a first-order phase transition from an imbalanced random state to structural balance with a critical temperature $T_c$, where in the case of $T>T_c$ there is no chance to reach the balanced state. A main feature of the first-order phase transition is the occurrence of a hysteresis loop crossing the balanced and imbalanced regimes.

Figure 1

Different types of triads according to structural balance.

Figure 2

(a) Graphical analysis of Eq. (10) for $N=50$ and in different temperatures $(T=15,28)$. When $T>T_c(T_c=26.2)$ there is a single stable fixed point. When $T<T_c$ there are three fixed points, and the inner one is unstable. The value of $T$ determines the number of fixed points. (b) Bifurcation diagram showing the fixed points ($q^{*}$) as a function of temperature ($T$).

Figure 3

The mean triad energy ($-\langle S_{ik}{S}_{kj}{S}_{ji}\rangle$) vs Monte Carlo steps ($n$) for different values of temperature in a fully connected network with $N=50$.

Figure 4

(a) Bifurcation diagram for the first-order phase transition in Eq. (11) as a function of temperature. The solid blue curve represents stable fixed points, while the dashed red curve is the unstable one. (b) Monte Carlo simulation from several initial states with different mean triad energies. To generate these initial states, we build a balanced network with all positive links and then randomly change the sign of some links to negative. There is a first-order phase transition at $T=T_c$, where the mean triad energy jumps from $-1$ to 0. This is known as a “blue-sky” bifurcation. The results correspond to a fully connected network with $N=50$.

Figure 5

The mean triad energy ($-\langle S_{ik}{S}_{kj}{S}_{ji}\rangle$) vs temperature ($T$) for different initial states differing in the percentage of balanced triads. We consider a balanced initial state with all positive links ($-\langle S_{ik}{S}_{kj}{S}_{ji}\rangle =-1$) and an imbalanced initial state with all negative links ($-\langle S_{ik}{S}_{kj}{S}_{ji}\rangle =1$). In a random initial state, we have almost equal positive and negative links ($-\langle S_{ik}{S}_{kj}{S}_{ji}\rangle \sim 0$).