Global graph curvature

Liudmila Prokhorenkova, Egor Samosvat, Pim van der Hoorn

TL;DR: Introduce a concept of global graph curvature specifically catered to the problem of embedding graphs and find its connection with popular local graph curvatures.

Abstract: Recently, non-Euclidean spaces became popular for embedding structured data. However, determining suitable geometry and, in particular, curvature for a given dataset is still an open problem. In this paper, we define a notion of global graph curvature, specifically catered to the problem of embedding graphs, and analyze the problem of estimating this curvature using only graph-based characteristics (without actual graph embedding). We show that optimal curvature essentially depends on dimensionality of the embedding space and loss function one aims to minimize via embedding. We review the existing notions of local curvature (e.g., Ollivier-Ricci curvature) and analyze their properties theoretically and empirically. In particular, we show that such curvatures are often unable to properly estimate the global one. Hence, we propose a new estimator of global graph curvature specifically designed for zero-one loss function.

Code: https://drive.google.com/open?id=12i5LD6yTyFRDrgkMJEencDAI0Au6Isrn

Keywords: graph curvature, graph embedding, hyperbolic space, distortion, Ollivier curvature, Forman curvature