Fig. 2: Hyper-core decomposition of empirical hypergraphs.
From: Hyper-cores promote localization and efficient seeding in higher-order processes
Panels a, e show colormaps giving the relative size n(k, m) (number of nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a function of k and m (white regions correspond to n(k, m) = 0). In the insets, n(k, m) is shown as a function of k at fixed values of m. Panels b, f show colormaps giving the z-score z(k, m) of the (k, m)-hyper-core relative size, with respect to 103 shuffled realizations of the hypergraph, as a function of k and m (values of z(k, m) ∈ (−1.96, 1.96) are shown in white). In panels c, g the size-independent hypercoreness R(i) is plotted as a function of the corresponding node rank; the insets give scatterplots of R(i) vs. the s-coreness, S(i), for all nodes. Panels d, h are the same as c, g, but for the frequency-based hypercoreness Rw(i). In panels a–d we consider the email-EU data set: R(i) and S(i) have a Pearson correlation coefficient of ρ = 0.90 (p-value p ≪ 0.001) and the corresponding rankings have a Kendall’s τ coefficient of τ = 0.85 (p ≪ 0.001), while Rw(i) and S(i) have ρ = 0.90 (p ≪ 0.001) and τ = 0.85 (p ≪ 0.001); in panels e–h we consider the music-review data set: R(i) and S(i) have ρ = 0.74 (p ≪ 0.001) and τ = 0.58 (p ≪ 0.001), while Rw(i) and S(i) have ρ = 0.98 (p ≪ 0.001) and τ = 0.89 (p ≪ 0.001).
