Measuring Partial Balance in Signed Networks

Is the enemy of an enemy necessarily a friend, or a friend of a friend a friend? If not, to what extent does this tend to hold? Such questions were formulated in terms of signed (social) networks and necessary and sufficient conditions for a network to be “balanced” were obtained around 1960. Since then the idea that signed networks tend over time to become more balanced has been widely used in several application areas. However investigation of this hypothesis has been complicated by the lack of a standard measure of partial balance, since complete balance is almost never achieved in practice. We formalize the concept of a measure of partial balance, discuss various measures, compare the measures on synthetic datasets, and investigate their axiomatic properties. The synthetic data involves Erdős-Rényi and specially structured random graphs. We show that some measures behave better than others in terms of axioms and ability to differentiate between graphs. We also use well-known datasets from the sociology literature, such as Read’s New Guinean tribes, and much more recent ones involving senate bill co-sponsorship. Our results on real social signed networks show that under various measures, they are much more balanced than what is expected by chance. We make some recommendations for measures to be used in future work.