Structural balance in fully signed networks

Abstract

Node attributes play an important role in shaping network structures, but are generally ignored in transformations of structural balance. A fully signed network consisting of signs of edges and nodes expresses both properties of relationship and node attributes. In this article, we generalize the definition of structural balance in fully signed networks. We transform the unbalanced fully signed network by not only changing signs of edges but also changing the signs of nodes. We propose a memetic algorithm to transform unbalanced networks at the lowest cost. Experiments show that our algorithm can solve this problem efficiently, and different node attribute assignments may lead to different optimized structures.

The evolution of cooperation in signed networks under the impact of structural balance

“…Heider first proposed the definition of structural balance to detect the origin of conflicts in networks [ 49 ], and this definition was generalized later by Cartwright and Harary based on graph theory [ 50 ]. Similarly to the evolution of cooperation based on the PD model, a series of studies has focused on the dynamic evolution of structural balance [ 51 – 57 ]. Du et al [ 57 ] analyzed structural balance in fully signed networks, where the node attribute could also be represented by signs “+” or “–”, and the network evolution was based on rules for simultaneously adjusting signs of relations and nodes.…”

“…Similarly to the evolution of cooperation based on the PD model, a series of studies has focused on the dynamic evolution of structural balance [ 51 – 57 ]. Du et al [ 57 ] analyzed structural balance in fully signed networks, where the node attribute could also be represented by signs “+” or “–”, and the network evolution was based on rules for simultaneously adjusting signs of relations and nodes. Some rules in this model can be also applied to the dynamic evolution of networks based on PD, because both models focus on the adjustment of nodes and relations.…”

“…According to the definition of structural balance in fully signed networks, there are only four kinds of balanced triangles, as shown in Fig 1 [ 57 ]. From a whole-network view, a balanced structure can be formed as shown in Fig 2 , which assumes the principle of homophily, namely, individuals tend to imitate their friends’ attributes, while having different attributes from those of their enemies [ 58 , 59 ].…”

“…According to Heider’s definition of structural balance [ 49 ], the balanced triangles consist of zero or two negative edges, which complies with the logic “my friend’s friend is my friend” and “my friend’s enemy is my enemy”. According to the structural balance in fully signed networks [ 57 ], nodes with same attributes should be connected by a positive edge, while pairs of nodes with opposite signs should be connected by a negative edge. Thus there are only four kinds of balanced triangles.…”

“…However, the triangles in Fig 3(B) and Fig 3(C) are not balanced according to the definition of structural balance in fully signed networks, because the negative relation connecting two defectors contributes to the global imbalance [ 57 ]. Moreover, under the impact of structural balance, the triangle in Fig 3(D) should be transformed to the form in Fig 1(D) , when the network becomes stable.…”