Title: Degree-Corrected Ricci Curvature for Networks
Primary area: learning on graphs and other geometries & topologies
Keywords: Network curvature, heterogeneity, community detection, asymptotics

Abstract: Networks are useful in understanding complex systems. Recently, discrete curvature measures, adapted from smooth manifolds to discrete landscape of networks, have been applied to network data analysis, such as community detection, with encouraging results. A fundamental hypothesis made for using these curvature measures is the diagonal-dominating principle: *the curvature measures are consistently larger within a community than between different communities.*
However, this principle may not hold under all statistical network models. 
We investigate three existing network curvatures, which satisfy the diagonal-dominating principle under the stochastic block model (SBM) but not under its widely-used degree-corrected version, the degree-corrected block model (DCBM). Observing that these curvature measures are heavily influenced by degree heterogeneity, we propose a new curvature measure, Degree-Corrected Ricci Curvature (DCRC), specifically designed to account for degree heterogeneity. Theoretically, we prove that DCRC always satisfies the diagonal-dominating principle under both SBM and DCBM. We also provide large-deviation bounds and uncertainty quantification. Empirically, we use DCRC to preprocess a network by filtering out low-curvature edges; and we show that this preprocessing step can improve the performance of state-of-art community detection algorithms.