Heider balance under disordered triadic interactions.

The Heider balance addresses three-body interactions with the assumption that triads are equally important in the dynamics of the network. In many networks, the relations do not have the same strength, so triads are differently weighted. Now, the question is how social networks evolve to reduce the number of unbalanced triangles when they are weighted? Are the results foreseeable based on what we have already learned from the unweighted balance? To find the solution, we consider a fully connected network in which triads are assigned with different random weights. Weights are coming from Gaussian probability distribution with mean μ and variance σ. We study this system in two regimes: (I) the ratio of μ/σ≥1 corresponds to weak disorder (small variance) that triads' weight are approximately the same; (II) μ/σ<1 counts for strong disorder (big variance) and weights are remarkably diverse. Investigating the structural evolution of such a network is our intention. We see disorder plays a key role in determining the critical temperature of the system. Using the mean-field method to present an analytic solution for the system represents that the system undergoes a first-order phase transition. For weak disorder, our simulation results display the system reaches the global minimum as temperature decreases, whereas for the second regime, due to the diversity of weights, the system does not manage to reach the global minimum.