Directed Acyclic Hypergraphs: Topology Identification from Nodal Data

Abstract

Inferring higher-order network structures from nodal data is an emerging challenge across fields such as signal processing, machine learning, and causal inference. While directed acyclic graphs (DAGs) provide a powerful framework for modeling causal or functional dependencies, they only capture pairwise interactions. This paper introduces a new directed acyclic hypergraph (DAH) signal model that generalizes DAG-based structural equation modeling to multi-node (higher-order) relationships. Our approach begins by lifting a directed hyper-graph (DH) into an equivalent bipartite directed graph (DG), where virtual nodes represent source-node sets of hyperedges. Nodal data are assigned to both original and virtual nodes, and an SEM is defined over the DG. By imposing a smooth acyclicity constraint on this bipartite graph, we obtain a continuous and scalable formulation for DAH estimation from nodal observations. The proposed framework unifies hypergraph learning and DAG inference under a common optimization perspective, enabling interpretable higher-order dependency discovery. Numerical experiments demonstrate the ability of the method to recover meaningful DAH structures from simulated nodal data.