Edgewise Envelopes Between Balanced Forman and Ollivier-Ricci Curvature

TL;DR

This paper introduces a scalable method to approximate Ollivier-Ricci curvature on large graphs by deriving explicit bounds using combinatorial local graph features, significantly reducing computational complexity.

Contribution

The authors develop explicit transfer bounds between OR and Forman curvature, enabling efficient, local graph-based estimation without global optimal transport computations.

Findings

Bounds accurately enclose empirical curvature distributions

Method reduces evaluation complexity to near-linear time

Analytical bounds are robust across diverse network types

Abstract

Evaluating Ollivier-Ricci (OR) curvature on large-scale graphs is computationally prohibitive due to the necessity of solving an optimal transport problem for every edge. We bypass this computational bottleneck by deriving explicit, two-sided, piecewise-affine transfer moduli between the transport-based OR curvature and the combinatorial Balanced Forman (BF) curvature introduced by Topping et al. By constructing a lazy transport envelope and augmenting the Jost and Liu bound with a cross-edge matching statistic, we establish deterministic bounds for $\mathfrak{c}_{OR}(i,j)$ parameterized by 2-hop local graph combinatorics. This formulation reduces the edgewise evaluation complexity from an optimal transport linear program to a worst-case $\mathcal{O}(\max_{v \in V} \operatorname{deg}(v)^{1.5})$ time, entirely eliminating the reliance on global solvers. We validate these bounds via distributional analyses on canonical random graphs and empirical networks; the derived analytical bands enclose the empirical distributions independent of degree heterogeneity, geometry, or clustering, providing a scalable, computationally efficient framework for statistical network analysis.