Fig. 3: Analytical reducibility of hyperrings.
a Eigenvalues of the multiorder Laplacians up to order d = 1, 2, for a hyperring with N = 100 nodes. b Eigenvalues of the corresponding density matrices, for τ = 0.01 (blue), 0.1 (orange), and 1 (green), for d = 1 (dashed) and d = 2 (solid). c Difference between the cost function at order d = 2 and d = 1 as a function of τ: it is negative (positive) when dopt = 2 (=1), i.e., the hypergraph is irreducible (reducible, beige shade). Vertical dashed lines indicate the short (\({\tau }_{{{{\rm{short}}}}}=1/{\lambda }_{\max }^{[2]}\)) and long (\({\tau }_{{{{\rm{long}}}}}=1/{\lambda }_{\min }^{[2]}\)) timescales of the full system.
