Summary: Researchers used tools from statistical physics and advanced network modeling to validate the long-standing axiom, “the enemy of my enemy is my friend.” Their work provides rigorous support for Fritz Heider’s social balance theory by incorporating realistic constraints—who knows whom and individual tendencies toward positivity—into null models for signed networks.
The new model more accurately reflects how real social networks form and evolve, and it clarifies why earlier attempts to test social balance produced mixed results. Beyond confirming a foundational social-science idea, the approach offers practical insight for studying and addressing complex phenomena such as political polarization, neural connectivity, and interactions between drugs.
Key facts
- Confirmation of Heider’s theory: By using a more realistic null model for signed networks, the study finds that social networks display the balanced relationship patterns Heider predicted.
- Improved network modeling: The researchers developed a model that preserves network topology and each node’s signed-degree preferences—accounting simultaneously for who is connected to whom and that some individuals tend to be more positive than others.
- Wide applicability: The modeling framework can be applied to diverse systems with positive and negative interactions, from social and political networks to neural circuits and drug-combination studies.
Context and motivation
The phrase “the enemy of my enemy is my friend” echoes a larger intuition about social harmony first formalized by Austrian psychologist Fritz Heider in the 1940s. Heider’s social balance theory describes how signed ties (friendly or hostile) among small groups tend toward configurations that minimize tension: for example, a friend of a friend is likely to be a friend, while a friend of an enemy tends to be an enemy.
Empirical tests of social balance have often yielded inconsistent or inconclusive results. Part of the difficulty lies in how researchers construct null models for comparison. Prior methods typically randomized edge signs without preserving essential features of real social systems, such as the sparse topology of who interacts with whom and the fact that some people are generally friendlier than others. Ignoring these constraints can mask genuine balance patterns or generate misleading outcomes.
What this study did
Researchers at Northwestern University developed a principled randomization method that enforces two realistic constraints simultaneously: the underlying network topology (who knows whom) and each node’s preference for positive or negative ties. Using a maximum-entropy framework, their STP null model preserves both structural constraints when randomizing signed networks. This properly calibrated null model allows statistically sound assessments of social balance in empirical data.
The team tested their approach on four large, publicly curated signed datasets: user ratings on Slashdot, recorded exchanges among members of the U.S. House of Representatives, trading interactions among Bitcoin users, and product reviews from Epinions. In contrast to previous randomizations, STP randomization distributes positive and negative signs conditioned on the observed topology and the signed-degree tendencies of individual nodes. The result: consistent evidence of strong balance patterns across three- and four-node structures and beyond.
Key results and interpretation
When both constraints are respected, large-scale social networks align with Heider’s expectations: balanced triads and larger balanced motifs appear more often than a properly constrained null model would predict. The findings indicate that earlier null models failed in part because they ignored signed-degree preferences and topology simultaneously. By matching these features, the STP model reveals the underlying tendency toward social balance that had been hidden by inadequate randomization procedures.
The study also shows that the principles of balance can extend beyond triads to larger network motifs, offering a potential wiring mechanism explaining how signed patterns emerge at multiple scales.
Implications and future directions
Beyond validating a core social theory, the STP framework opens several applied avenues. One promising direction is exploring interventions aimed at reducing political polarization by identifying structural and attitudinal constraints that sustain hostile divisions. More broadly, the same modeling approach can be adapted to systems with excitatory and inhibitory interactions—such as neuronal networks—or to analyze combinations of drugs where interactions can be beneficial or detrimental.
“We always believed the social intuition was sound, but we needed the correct mathematics to test it,” said István Kovács, the study’s senior author. “By incorporating who knows whom and individuals’ overall friendliness simultaneously, we can now demonstrate that social networks align with expectations formed 80 years ago.”
Bingjie Hao, the study’s first author, added that the mathematics behind the model is straightforward yet powerful, and that the framework will be useful for modeling interactions in many complex systems beyond social groups.
About this physics and social neuroscience research news
Author: Amanda Morris
Source: Northwestern University
Contact: Amanda Morris – Northwestern University
Image: The image is credited to Neuroscience News
Original research (open access): “Proper network randomization is key to assessing social balance” by István Kovács et al., published in Science Advances.
Abstract
Proper network randomization is key to assessing social balance
Signed ties—positive or negative—create specific patterns in networks that balance theory seeks to describe. Strong balance generates cycles with even numbers of negative edges and other characteristic motifs. The statistical significance of these patterns is typically evaluated against null models, but current null models can fail even when a network is balanced by design.
This work demonstrates that matching nodes’ signed-degree preferences and preserving network topology are both critical steps for reliable null models. The proposed STP null model integrates these constraints within a maximum-entropy framework. STP randomization yields qualitatively different conclusions: most empirical social networks consistently show strong balance across three- and four-node structures. Based on these results, the authors outline a possible wiring mechanism behind observed signed patterns and discuss further applications of STP randomization to other complex systems.