2024/12/07 by Anna C. Gilbert, Gilbert, Anna C., Joon-Hyeok Yim +1
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Mathematical Dynamics and Fractals #Morphological variations and asymmetry
paper · pdf · doi:10.48550/arxiv.2412.05746
A metric space (X,d) is said to be δ-hyperbolic if d(x,y)+d(z,w) is at most max(d(x,z)+d(y,w), d(x,w)+d(y,z)) by 2 δ. A geodesic space is δ-slim if every geodesic triangle Δ(x,y,z) is δ-slim. It is well-established that the notions of δ-slimness, δ-hyperbolicity, δ-thinness and similar concepts are equivalent up to a constant factor. In this paper, we investigate these properties under an average-case framework and reveal a surprising discrepancy: while 𝔼δ-slimness implies 𝔼δ-hyperbolicity, the converse does not hold. Furthermore, similar asymmetries emerge for other definitions when comparing average-case and worst-case formulations of hyperbolicity. We exploit these differences to analyze the random Gaussian distribution in Euclidean space, random d-regular graph, and the random Erdős-Rényi graph model, illustrating the implications of these average-case deviations.