Physics Confirms the Age-Old Axiom: "The Enemy of Your Enemy is Your Friend"

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Most people have heard the adage, "the enemy of my enemy is my friend". This saying encapsulates a social intuition that people have relied on for generations. Now, researchers at Northwestern University have used statistical physics to confirm that this axiom holds true in real-world social networks, lending scientific credibility to the theory behind it.

In the 1940s, Austrian psychologist Fritz Heider introduced the concept of social balance theory, positing that humans are innately driven to seek harmony within their social circles. According to Heider's theory, four rules create balanced relationships:

The enemy of an enemy is a friend.
The friend of a friend is a friend.
The friend of an enemy is an enemy.
The enemy of a friend is an enemy.

When these rules are followed, relationships tend to be harmonious. However, if any of these rules are broken, it can lead to imbalances, causing tension and conflict within social networks.

Social relations. Credit: István Kovács / Northwestern University

Over the years, many studies have tried to validate Heider's theory using network science and mathematical models. However, these efforts often fell short because they assumed idealized or oversimplified networks that did not accurately capture the complexities of human relationships. In reality, not everyone knows everyone else, and people differ in their natural inclination toward friendliness. These gaps in the models led to inconsistent results and cast doubt on whether Heider's theory truly described real-world social dynamics.

The Northwestern team, led by István Kovács, overcame these limitations by developing a new network model that incorporates two key factors: the limited connectivity of real-world networks and varying degrees of individual friendliness. By accounting for these factors simultaneously, the researchers created a more realistic representation of social networks and confirmed Heider's theory with much greater accuracy.

Their work not only confirms that the enemy of your enemy is, indeed, your friend, but also provides a framework that can be used to understand broader social dynamics. The team's model allows researchers to explore phenomena like political polarization and international relations, where balanced and unbalanced relationships can have significant implications.

(A) Signed node copying. Node A′ copies the edges and signs from node A, forming balanced squares except the + − + − square. (B) Edge copying. Existing connected nodes copy each other's edges with preserved signs (if positively connected) or reversed signs (if negatively connected). (C) Edge copying with node addition. New nodes are added to the network and copy all or some of the edges from a connected node following the edge-copying rules. This process eventually forms triangles, squares (including the + − + − square B − C − D − E − B), squareZs, and squareXs. Blue lines indicate positive edges and red lines indicate negative edges. Copied edges are shown in dashed lines. Credit: I.A. Kovács et al. / ScienceAdvances

The implications of this new framework extend beyond social relationships. The model's ability to handle complex interactions with varying constraints could be applied to other systems where positive and negative interactions play a role. This includes neural networks in the brain, drug interactions, and even climate systems.

The code and data used for the study are publicly available, offering researchers a valuable resource for further investigation. The Northwestern team's work represents a significant step forward in understanding the underlying mechanics of social harmony and discord.

Kovács and his team are already exploring new avenues for this research. They are interested in applying their model to study potential interventions that could reduce political polarization, a pressing issue in today's society. Additionally, the model could be used to study a wide range of complex networks, offering insights into how different systems operate and interact.

As Kovács noted, the key to this discovery was taking a realistic approach to modeling social networks. By acknowledging the complexities and embracing mathematical rigor, the team has paved the way for a deeper exploration of human relationships and beyond.

Sources

Northwestern University | Bingjie Hao, István A. Kovács , Proper network randomization is key to assessing social balance. Sci. Adv. 10, eadj0104(2024). DOI:10.1126/sciadv.adj0104

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