  Despite the widespread adoption of higher-order mathematical structures such
as hypergraphs, methodological tools for their analysis lag behind those for
traditional graphs. This work addresses a critical gap in this context by
proposing two micro-canonical random null models for directed hypergraphs: the
Directed Hypergraph Configuration Model (DHCM) and the Directed Hypergraph
JOINT Model (DHJM). These models preserve essential structural properties of
directed hypergraphs such as node in- and out-degree sequences and hyperedge
head and tail size sequences, or their joint tensor. We also describe two
efficient MCMC algorithms, NuDHy-Degs and NuDHy-JOINT, to sample random
hypergraphs from these ensembles.
  To showcase the interdisciplinary applicability of the proposed null models,
we present three distinct use cases in sociology, epidemiology, and economics.
First, we reveal the oscillatory behavior of increased homophily in opposition
parties in the US Congress over a 40-year span, emphasizing the role of
higher-order structures in quantifying political group homophily. Second, we
investigate non-linear contagion in contact hyper-networks, demonstrating that
disparities between simulations and theoretical predictions can be explained by
considering higher-order joint degree distributions. Last, we examine the
economic complexity of countries in the global trade network, showing that
local network properties preserved by NuDHy explain the main structural
economic complexity indexes.
  This work advances the development of null models for directed hypergraphs,
addressing the intricate challenges posed by their complex entity relations,
and providing a versatile suite of tools for researchers across various
domains.
