Panel a depicts a categorical system comprised of a set four categories (equivalently “elements” or “partitions”) connected by undirected edges whose lengths are proportional to the distance between categories. In this case, the distances between categories are all equal (symmetric), and the within-category distance is 0 (as evident in the depicted distance matrix). These properties define the set as categorical. The size of the nodes represents their relative abundance, which is also shown in the corresponding bar chart. Panel b demonstrates samples from nine categorical systems with varying number of categories (2, 3, and 4) and varying levels of inequality in the abundance distribution. Systems in the upper row have the highest level of inequality in abundance, whereas the systems shown in the bottom row have perfectly even abundance distributions. Together, these plots demonstrate that heterogeneity increases with both (A) increases in the number of categories and (B) more evenly distributed abundance across categories.
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