  Most of the current hypergraph learning methodologies and benchmarking
datasets in the hypergraph realm are obtained by lifting procedures from their
graph analogs, leading to overshadowing specific characteristics of
hypergraphs. This paper attempts to confront some pending questions in that
regard: Q1 Can the concept of homophily play a crucial role in Hypergraph
Neural Networks (HNNs)? Q2 Is there room for improving current HNN
architectures by carefully addressing specific characteristics of higher-order
networks? Q3 Do existing datasets provide a meaningful benchmark for HNNs? To
address them, we first introduce a novel conceptualization of homophily in
higher-order networks based on a Message Passing (MP) scheme, unifying both the
analytical examination and the modeling of higher-order networks. Further, we
investigate some natural, yet mostly unexplored, strategies for processing
higher-order structures within HNNs such as keeping hyperedge-dependent node
representations, or performing node/hyperedge stochastic samplings, leading us
to the most general MP formulation up to date -MultiSet-, as well as to an
original architecture design, MultiSetMixer. Finally, we conduct an extensive
set of experiments that contextualize our proposals and successfully provide
insights about our inquiries.
