Heider Balance under Disordered Triadic Interactions

TL;DR

This paper investigates how weighted triadic interactions influence the evolution of social networks towards balance, revealing phase transitions and a pseudo-Kondo effect depending on the level of disorder in weights.

Contribution

It introduces a weighted Heider balance model with Gaussian-distributed triad weights and analyzes its phase transition behavior under different disorder regimes.

Findings

Weak disorder leads to global minimum at low temperatures.

Strong disorder results in a pseudo-Kondo effect and prevents reaching global minimum.

Disorder significantly affects the critical temperature of the system.

Abstract

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 $\mu$ and variance $\sigma$. We study this system in two regimes : (\RN{1}) the ratio of $\frac{\mu}{\sigma} \ge 1 $ corresponds to weak disorder (small variance) that triads' weight are approximately the same, (\RN{2}) $\frac{\mu}{\sigma} < 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 high disorder we manage to see a seudo-Kondo effect in triadic interactions. The Kondo effect explains the anomalous increase in electrical resistivity of metals due to magnetic impurities at low temperatures. Similar behavior is seen in the energy versus temperature in our work for the second regime and the system does not reach a global minimum.