Higher-Order Null Models as a Lens for Social Systems

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 microcanonical random null models for directed hypergraphs: the directed hypergraph degree model (dhdm) 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 Markov chain Monte Carlo 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 U.S. Congress over a 40-year span, emphasizing the role of higher-order structures in quantifying political group homophily. Second, we investigate a nonlinear contagion in contact hypernetworks, 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.