Individual versus social benefit on heterogeneous networks.

Abstract

The focus of structural balance theory is dedicated to social benefits, while in a real network individual benefits sometimes are the focus as well. The Strauss's model addresses individual benefits besides the social one with a simple assumption that all individual benefits are equivalent. Therefore, the results show that the competition between two terms leads to a phase transition between individual and social benefits and there is a critical point CP that represents a first-order phase transition in the network. Concerning a real network of relations, individuals adjust the strength of their relationships based on the benefits they acquire from them. Hence, by addressing heterogeneity in the individual interactions, we study a modified version of Strauss's model in which the first term represents the heterogeneous individual benefits by θ_{ij}, and the coefficient of the second term, α, measures the strength of social benefits. Our studies show that there is a region where the triangles are in a crumpled state rather than being dispersed in the network; furthermore, increasing the heterogeneity of individual benefits results in the narrower region of the crumpled state. Outside of the mentioned region, the network is a mixture of links and triangles and the value of α determines whether the individual benefit or social benefit dominates. For the small value of α, the individual benefit dominates, whereas in the large value of α, the social benefit dominates.