  Hypergraphs provide a natural way of representing group relations, whose
complexity motivates an extensive array of prior work to adopt some form of
abstraction and simplification of higher-order interactions. However, the
following question has yet to be addressed: How much abstraction of group
interactions is sufficient in solving a hypergraph task, and how different such
results become across datasets? This question, if properly answered, provides a
useful engineering guideline on how to trade off between complexity and
accuracy of solving a downstream task. To this end, we propose a method of
incrementally representing group interactions using a notion of n-projected
graph whose accumulation contains information on up to n-way interactions, and
quantify the accuracy of solving a task as n grows for various datasets. As a
downstream task, we consider hyperedge prediction, an extension of link
prediction, which is a canonical task for evaluating graph models. Through
experiments on 15 real-world datasets, we draw the following messages: (a)
Diminishing returns: small n is enough to achieve accuracy comparable with
near-perfect approximations, (b) Troubleshooter: as the task becomes more
challenging, larger n brings more benefit, and (c) Irreducibility: datasets
whose pairwise interactions do not tell much about higher-order interactions
lose much accuracy when reduced to pairwise abstractions.
