Structural Balance Considerations for Networks with Preference Orders as Node Attributes

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.