The enemy of your enemy is really your friend – according to physics!

Humans are social animals whose web of relationships is complex and often changing. Understanding these social networks and their changes has been the task of different sciences and different theories. One of these was proposed in the 1940s, the so-called social equilibrium theory. Now researchers have been able to confirm this using statistical physics.

The idea at the heart of social equilibrium theory is, as the name suggests, equilibrium. Individuals want and try to maintain balanced relationships within their networks. Positive relationships are balanced, while negative or mixed relationships are not. You need rules to keep the system in balance. The classical model has four simple rules, based on the simplistic idea that positive relationships are “friends” and negative relationships are “enemies.”

The first rule is that a friend of a friend is a friend. This is an idealized case: don’t immediately think of that friend of your friend who hates you. Another rule is that a friend of an enemy is an enemy, and of course the enemy of a friend is an enemy: we must defend our friends. The last rule is more subtle: an enemy of an enemy is a friend. The new analysis appears to be consistent with this requirement, but the scientists had to introduce significant complexity before they could model it.

“We can finally conclude that social networks match the expectations formed 80 years ago,” said first author Bingjie Hao of Northwestern University in a statement. “Our findings also have broad applications for future use. Our math allows us to include constraints on the connections and preferences of different entities in the system. That will be useful for modeling other systems outside social networks.”

Crucial to the new model were two factors: not everyone knows everyone in real life, and some people are more positive than others. Using both constraints essentially reproduces a social network like the one Fritz Heider predicted 80 years ago.

“We always thought this social intuition works, but we didn’t know why it worked,” said István Kovács, lead author of the study. “All we needed was to figure out the math. If you look through the literature, there are many studies on the theory, but there is no agreement between them. For decades we have been wrong all the time. The reason is that real life is complicated. We realized that we had to take into account both limitations at the same time: who knows who and that some people are just friendlier than others.”

The study was published in the journal Science Advances.