Abstract
This paper investigates the spectral properties of normalized Laplacian matrices defined on directed weighted hypergraphs and explores their direct applications to multi-agent consensus protocols. Unlike traditional graphs that only model pairwise interactions, hypergraphs allow for the representation of multi-body relationships where hyperedges connect arbitrary subsets of vertices. By incorporating both directionality and edge/vertex weighting schemes, we establish a generalized mathematical framework for directed weighted hypergraphs. We define a novel normalized Laplacian matrix for these structures and rigorously analyze its spectral properties, including eigenvalue bounds, the existence of zero eigenvalues, and the localization of the spectrum on the complex plane using Gerschgorin's circle theorem. Furthermore, we apply these spectral insights to both continuous-time and discrete-time consensus protocols. We prove that the algebraic connectivity, characterized by the spectral gap of the normalized Laplacian, dictates the asymptotic convergence rate of the consensus protocol. Finally, numerical simulations are presented to validate our theoretical results, illustrating how hypergraph topologies influence consensus dynamics compared to traditional pairwise graph models.
