Sensitivity of Collective Outcomes Identifies Pivotal Components

Abstract

A social system is susceptible to perturbation when its collective properties depend sensitively on a few, pivotal components. Using the information geometry of minimal models from statistical physics, we develop an approach to identify pivotal components to which coarse-grained, or aggregate, properties are sensitive. As an example we introduce our approach on a reduced toy model with a median voter who always votes in the majority. With this example, we construct the Fisher information matrix with respect to the distribution of majority-minority divisions and study features of the matrix that pinpoint the unique role of the median. More generally, these features identify pivotal blocs that precisely determine collective outcomes generated by a complex network of interactions. Applying our approach to data sets from political voting, finance, and Twitter, we find remarkable variety from systems dominated by a median-like component (e.g., California State Assembly) to those without any single special component (e.g., Alaskan Supreme Court). Other systems (e.g., S&P sector indices) show varying levels of heterogeneity in between these extremes. By providing insight into such sensitivity, our information-geometric approach presents a quantitative framework for considering how nominees might change a judicial bench, serve as a measure of notable temporal variation in financial indices, or help analyze the robustness of institutions to targeted perturbation.