Mathematical Foundations of Data Cohesion

Abstract: Data cohesion, a recently introduced measure inspired by social interactions, uses distance comparisons to assess relative proximity. In this work, we provide a collection of results which can guide the development of cohesion-based methods in exploratory data analysis and human-aided computation. Here, we observe the important role of highly clustered "point-like" sets and the ways in which cohesion allows such sets to take on qualities of a single weighted point. In doing so, we see how cohesion complements metric-adjacent measures of dissimilarity and responds to local density. We conclude by proving that cohesion is the unique function with (i) average value equal to one-half and (ii) the property that the influence of an outlier is proportional to its mass. Properties of cohesion are illustrated with examples throughout.