Modeling Small Systems Through the Relative Entropy Lattice

There are certain contexts, where we would like to analyze the behavior of small interacting systems, such as sports teams. While large interacting systems have drawn much attention in the past years, let it be physical systems of interacting particles or social networks, small systems are short of appropriate quantitative modeling and measurement tools. We propose a simple procedure for analyzing a small system through the degree in which its behavior at different granularity levels (e.g., dyads) non-linearly diverges from the simple additive behavior of its sub-units. For example, we may model the behavior of a soccer team by measuring the extent to which the behavior changes when we move from individual players to dyads, triads, and so on. In this paper, we address the challenge of modeling small systems in terms of measuring divergence from additivity at different granularity levels of the system. We present and develop a measure for quantifying divergence from additivity through what we term a Relative Entropy Lattice, and illustrate its benefits in modeling the behavior of a specific small system, a soccer team, using data from the English Premier League. Our method has practical implications too, such as allowing the coach to identify 'hidden' weak spots in the team's behavior.