  The Adjusted Rand Index (ARI) is a widely used method for comparing hard
clusterings, but requires a choice of random model that is often left implicit.
Several recent works have extended the Rand Index to fuzzy clusterings, but the
assumptions of the most common random model is difficult to justify in fuzzy
settings. We propose a single framework for computing the ARI with three random
models that are intuitive and explainable for both hard and fuzzy clusterings,
along with the benefit of lower computational complexity. The theory and
assumptions of the proposed models are contrasted with the existing permutation
model. Computations on synthetic and benchmark data show that each model has
distinct behaviour, meaning that accurate model selection is important for the
reliability of results.
