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.

翻译：在大规模图上计算Ollivier-Ricci（OR）曲率需要为每条边求解最优传输问题，因而在计算上难以实现。我们通过推导基于传输的OR曲率与Topping等人提出的组合平衡型Forman（BF）曲率之间显式、双边、分段仿射传输模量，绕过了这一计算瓶颈。通过构建惰性传输包络，并引入跨边匹配统计量对Jost-Liu界进行增强，我们建立了由2跳局部图组合结构参数化的$\mathfrak{c}_{OR}(i,j)$确定性界。该公式将边评估复杂度从最优传输线性规划降低至最坏情况$\mathcal{O}(\max_{v \in V} \operatorname{deg}(v)^{1.5})$时间，完全消除了对全局求解器的依赖。我们通过在典型随机图与经验网络上的分布分析验证了这些界；所得解析带独立于度异质性、几何结构或聚类特征，始终包含经验分布，为统计网络分析提供了可扩展、计算高效的框架。