Edgewise Envelopes Between Balanced Forman and Ollivier-Ricci Curvature

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.

Figures (5)

• Figure 1: Local, small-multiple illustrations around a fixed base edge $e=(u,v)$. (a) $\triangle(u,v)=|\mathcal{N}(u)\cap\mathcal{N}(v)|$ is visualized by common neighbors (two shown). (b) $\Xi_{uv}$ counts vertices in $\mathcal{U}_u\cup\mathcal{U}_v$ that are incident to at least one cross edge: the dashed cross edges (with ring-marked vertices) are counted. (c) For a chosen $k\in\mathcal{N}(u)\setminus\{v\}$, dotted edges indicate contributions to $\widetilde{\Box}(k,u,v)$; maximizing over $k$ shows the contribution of $\varpi_{\max}(u,v)$.
• Figure 2: Toy two-block stochastic block model (SBM) graphs illustrating assortative and disassortative regimes. Dotted edges denote within-block connections, while solid edges denote between-block connections. In (a), the assortative regime ($p_{\rm in}\gg p_{\rm out}$) yields predominantly within-block connectivity with only a few cross-block edges. In (b), the disassortative regime ($p_{\rm out}\gg p_{\rm in}$) produces predominantly cross-block connectivity.
• Figure 3: Representative edgewise scatter plots. Blue points: sampled edges; black: $\mathrm{median}[\mathfrak c_{\rm OR}\mid \mathfrak c_{\rm BF}]$; green (red) dash-dot: median of the lower (upper) transfer $\mathfrak c_{\rm BF}\mapsto\mathfrak{c}_{\rm OR}$. Panels: RGG, ER, WS, BA (left-to-right, top-to-bottom).
• Figure 4: Representative edgewise scatter plots. Same styling as Figure \ref{['fig:scatters_A']}. Panels: HRG, RGG, SBM (assortative), SBM (disassortative).
• Figure 5: Distributional views.Top row (RGG): observed histograms (filled blue) against distributions induced by transfers (green/orange/red outlines) and by the coverage envelope (Proposition \ref{['prop:coverage-envelope-monotone']}). Middle row (ER): sparse case where the lower distributions track the observed modes. Bottom row (WS): quantile profiles for observed series and their transferred counterparts, showing near-parallel separation and a visible "knee" at low quantiles produced by partially rewired neighborhoods.

Theorems & Definitions (42)

• Remark 2.1
• Definition 2.2: Walks and Paths
• Definition 2.3: Graph Distance
• Remark 2.4
• Definition 2.5: Lazy Random-Walk Measure
• Remark 2.6: Uniform Mass on the Closed Neighborhood
• Definition 2.7: Couplings and Transport Plans
• Remark 2.8
• Definition 2.9: $W_1$ Distance on a Graph
• Lemma 2.10: Existence of Optimal Couplings on Finite Graphs