A random matrix perspective of cultural structure: groups or redundancies?

Abstract

Abstract Recent studies have highlighted interesting properties of empirical cultural states—collections of cultural trait sequences of real individuals. Matrices of similarity between individuals may be constructed from these states, allowing for more insights to be gained using random matrix techniques, approach first exploited in this study. We propose a null model that enforces, on average, the empirical occurrence frequency of each possible trait. With respect to this null model, the empirical matrices show deviating eigenvalues, which may be signatures of subtle cultural groups. However, they can conceivably also be artifacts of arbitrary redundancies between cultural variables. We study this possibility in a highly simplified setting, allowing for a side-by-side mathematical comparison of the two scenarios (groups and redundancies). The scenarios are shown to be completely indistinguishable in terms of deviating eigenvalues, confirming that the latter can in general be signatures of either redundancies or groups. The scenarios can be distinguished after evaluating the eigenvector uniformities and the associated deviations from null model expectations. This provides a uniformity-based validation criterion, which is reliable when searching for groups that are internally uniform, but fails when these exhibit significant internal non-uniformity. For empirical data, all the relevant eigenvector uniformities are compatible with the null model, indicating the absence of any internally uniform groups. Although there are various indications that some of the deviating eigenvalues could correspond to internally non-uniform groups, a generic procedure for distinguishing such groups from redundancy artifacts requires further research.