Individual versus Social Benefit on the Heterogeneous Networks

TL;DR

This paper investigates how heterogeneity in individual benefits influences the balance between individual and social benefits in networks, revealing a phase transition and the conditions leading to different network states.

Contribution

It introduces a modified Strauss's model incorporating heterogeneous individual benefits and analyzes how this heterogeneity affects network phase transitions and structure.

Findings

Heterogeneity narrows the crumpled state region in the network.

A critical point marks a first-order phase transition between benefit dominance.

Network structure varies with the strength of social versus individual benefits.

Abstract

The focus of structural balance theory is dedicated to social benefits, while in a real network individual benefits sometimes get the importance as well. In Strauss's model, the local minima are modeled by considering an individual term besides a social one and the assumption is based on equal strength of individual benefits. 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. Therefore, addressing heterogeneity in the individual interactions, we study a modified version of Strauss's model in which the first term represents the heterogeneous individual benefit by $\theta_{ij}$, and the coefficient of the second term, $\alpha$, measures the strength of social benefit. Our studies show that there is a region where the triangles are in a crumpled state rather than being dispersed in the network and increasing the heterogeneity of individual benefits results in the narrower region of crumpled state. Out of this region, the network is a mixture of links and triangles and the value of $\alpha$ determines whether the individual benefit or social benefit overcomes. For the small value of $\alpha$ the individual benefit dominates whereas in the large value of $\alpha$ the social benefit overcomes.