Structural Balance Considerations for Networks with Preference Orders as Node Attributes

TL;DR

This paper extends structural balance theory to networks with preference orderings as node attributes, classifies triangle configurations into equivalence classes, and compares empirical data to random preference models.

Contribution

It introduces a classification of preference-based triangle configurations into equivalence classes and derives a general formula for any number of alternatives.

Findings

Reduced 216 triangle configurations to 10 classes for 3 alternatives

Derived a formula for the number of classes with n alternatives

Empirical data shows deviations from random preference distribution

Abstract

We discuss possible definitions of structural balance conditions in a network with preference orderings as node attributes. The main result is that for the case with three alternatives ($A,B,C$) we reduce the $(3!)^3 = 216$ possible configurations of triangles to $10$ equivalence classes, and use these as measures of balance of a triangle towards possible extensions of structural balance theory. Moreover, we derive a general formula for the number of equivalent classes for preferences on $n$ alternatives. Finally, we analyze a real-world data set and compare its empirical distribution of triangle equivalence classes to a null hypothesis in which preferences are randomly assigned to the nodes.