Augmented Forman Ricci Curvature Overview

• AFRC is a discrete curvature invariant that extends Forman–Ricci curvature by incorporating higher-dimensional simplicial structures such as triangles, quadrangles, and pentagons.
• It provides an edge-centric, closed-form curvature measure computable in linear time with strong empirical correlation to Ollivier–Ricci curvature in sparse networks.
• AFRC is applied in community detection, GNN rewiring, and discrete geometry, offering robust insights for network analysis and resilience assessment.

@@@@2@@@@ Forman Ricci Curvature (AFRC) is a discrete curvature invariant for (possibly weighted and directed) networks. It extends the original Forman–Ricci curvature—defined on the 1-skeleton of CW-complexes—by incorporating higher-dimensional simplicial structure, notably small cycles (triangles, quadrangles, etc.), as higher-order faces. AFRC produces a closed-form, combinatorial, and edge-centric curvature measure computable in linear (or near-linear) time and displays strong empirical and theoretical correlation with Ollivier–Ricci curvature, particularly in sparse and real-world network scenarios. Beyond its geometric significance, AFRC supports a range of applications in network analysis, including community detection and curvature-driven graph rewiring.

1. Mathematical Foundations and Definitions

The original Forman–Ricci curvature for an edge $e = (v_1,v_2)$ in an unweighted, undirected graph $G=(V,E)$ is

$\mathcal{F}(e) = 4 - \deg(v_1) - \deg(v_2)$

which assesses local “dispersal” via endpoint degrees (Fesser et al., 2023).

AFRC augments this by filling in small cycles of length $n$ (3 for triangles, 4 for quadrangles, 5 for pentagons, etc.) with higher-dimensional faces and incorporates these in the curvature:

General formula (unweighted, quasiconvex case):

$$\mathcal{F}_{\text{Aug}}(e) = \mathcal{F}(e) + \sum_{n\geq 3} (6-n)\cdot |f^n_e|$$

where $|f^n_e|$ counts the number of $n$–gonal faces (simple $n$–cycles) containing $e$ (Iváñez, 2022).

Explicit cycle truncations frequently considered:

• Triangle-augmented:

$$\mathcal{F}_\triangle(e) = \mathcal{F}(e) + 3\,|f^3_e|$$

• Up to pentagons:

$\mathcal{F}_\pentagon(e) = \mathcal{F}(e) + 3\,|f^3_e| + 2\,|f^4_e| + 1\,|f^5_e|$

In directed networks, Samal et al. define for an edge $e_{1\to 2}$ with tail $v_1$ and head $v_2$:

• $E_{I,v_1}$: incoming edges to $v_1$ (excluding $e$)
• $E_{O,v_2}$: outgoing edges from $v_2$ (excluding $e$)
• $T(e)$: set of directed triangles (feed-forward loops) containing $e$ with consistent orientation

Directed AFRC:

$$F^\#(e) = w_e \left[ \sum_{t\in T(e)} \frac{w_e}{w_t} + \frac{w_{v_1}}{w_e} + \frac{w_{v_2}}{w_e} - \sum_{f\in E_{I,v_1}} \frac{w_{v_1}}{\sqrt{w_e w_f}} - \sum_{g\in E_{O,v_2}} \frac{w_{v_2}}{\sqrt{w_e w_g}} \right]$$

For unweighted graphs, this reduces to

$$F^\#(e) = |T(e)| + 2 - |E_{I,v_1}| - |E_{O,v_2}| = F(e) + |T(e)|$$

where $F(e) = 2 - \text{indeg}(v_1) - \text{outdeg}(v_2)$ (Saucan et al., 2018).

In non-quasiconvex networks where faces may overlap on multiple edges, orientation-sensitive counts are required to avoid overcounting (Iváñez, 2022).

2. Geometric and Structural Interpretation

AFRC generalizes the discrete Bochner–Weitzenböck formalism to encode not only local star structure (endpoint degrees), but also mesoscopic motifs and higher-order connectivity. Each short closed cycle (triangle, square, pentagon, etc.) including $e$ increases its AFRC:

• Triangles (3-cycles): contribute $+3$
• Quadrangles (4-cycles): contribute $+2$
• Pentagons (5-cycles): contribute $+1$
• Longer cycles: contribute $6-n$

This quantifies local “filling in” of holes, flattening angle deficits, and thus provides a combinatorial analogue of Gauss–Bonnet corrections. In directed graphs, only cycles of consistent (feed-forward) orientation are included, reflecting motif directionality (Saucan et al., 2018).

AFRC thus directly senses the density and configuration of small cycles at the edge level, unlike plain Forman–Ricci, which is fully determined by endpoint degrees, or Ollivier–Ricci, which measures optimal-transport between neighborhood distributions.

3. Computational Properties and Algorithmic Implementation

AFRC is computable purely locally:

• Degree computation: $O(|E|)$
• Triangle count per edge: $O(\min(\deg(u),\deg(v)))$ in adjacency-list representation
• Extension to $k$-gon faces (cycle length $k$) scales as $O(|E|\,d^{k-2})$ for average degree $d$ (Samal et al., 2017, Iváñez, 2022)

Directed AFRC for each edge requires intersecting relevant neighbor lists to count feed-forward triangles, with overall complexity $O(m d)$ for $m=|E|$, average degree $d$ (Saucan et al., 2018).

This is dramatically faster than the computation of Ollivier–Ricci curvature, which for each edge involves an $O(d^3)$ (Hungarian algorithm) or $O(d^2)$ (Sinkhorn approximation) optimal transport subroutine. AFRC thus enables efficient curvature computation even for million-edge networks (Fesser et al., 2023, Fesser et al., 2023).

In practice, restricting augmentation to triangle counts ($k=3$) is advisable for large and non-quasiconvex networks, as higher-cycle counts may be distorted by overlaps (Iváñez, 2022).

4. Empirical Properties and Correlations With Other Curvature Notions

Multiple studies confirm strong Spearman and Pearson correlations (typically $0.7$–$0.9$ in sparse networks) between AFRC and Ollivier–Ricci curvature at the edge level across model (Erdős–Rényi, Watts–Strogatz, Barabási–Albert, hyperbolic random graphs) and real networks (power grid, email, PGP, protein interaction networks) (Samal et al., 2017, Fesser et al., 2023).

In directed or undirected cases, this high correspondence is especially pronounced when augmenting plain Forman–Ricci curvature by triangle contributions, versus non-augmented Forman–Ricci, which can fail to track network “curvedness” in motifs-rich regimes (Saucan et al., 2018).

Under quasi-convex or “independent short-cycle” conditions, AFRC with faces up to pentagons matches mean-field Ollivier–Ricci curvature up to a degree-rescaling factor, recovering exact equivalence in certain regimes (Tee et al., 2021).

Empirically, AFRC distributions in model networks reflect known modular structure:

• Edges inside communities: higher (more positive) AFRC
• Edges between communities: lower (often negative) AFRC
• In real-data, AFRC strongly marks bottleneck or backbone connections, e.g., the $\sigma$–factor “core” in E. coli transcriptional regulatory networks (Saucan et al., 2018)

5. Applications in Network Analysis and Learning

AFRC has been successfully applied across diverse domains:

• Community Detection: Edges with high AFRC cluster inside communities, while inter-community edges have lower curvature. Algorithmic frameworks use sequential edge deletion based on AFRC, with Gaussian mixture thresholding, to segment communities with accuracy and curvature-gap metrics comparable to Ollivier–Ricci–based approaches, yet with order-of-magnitude speedups (Fesser et al., 2023).
• GNN Rewiring and Over-squashing/Over-smoothing Mitigation: AFRC quantifies bottlenecks (low curvature: over-squashing) and edges prone to over-smoothing (high curvature). Rewiring based on AFRC—adding edges where AFRC is low, pruning where it is high—mitigates both phenomena and improves classification accuracy in GNNs, with scalability absent in ORC-based rewiring (Fesser et al., 2023).
• Robustness and Resilience: Very negative AFRC edges correspond to bottlenecks: their removal rapidly disintegrates network efficiency (Saucan et al., 2018).
• Discrete Geometry and Quantum Gravity: In combinatorial quantum gravity models, enhanced (pentagon-augmented) Forman curvature matches Ollivier–Ricci in pre-geometric regimes and tracks emergence of flat or positively curved phases via local cycle condensation (Tee et al., 2021).
• Persistent Homology: Triangle/face-augmented AFRC provides stable, interpretable filtrations for persistent homology, capturing essential cycle structure (Iváñez, 2022).

6. Theoretical Relationships and Regimes of Validity

AFRC subsumes plain Forman–Ricci for trivial cycle structure and converges to Ollivier–Ricci in large-degree or quasi-convex configurations:

• For unweighted undirected graphs (with no faces): AFRC reduces to $4 - \deg(v_1) - \deg(v_2)$.
• Including only triangles ($k=3$): AFRC simplifies to $4 - \deg(v_1) - \deg(v_2) + 3 \times$[triangle count].
• For pentagon- or higher-augmentation, AFRC approaches Ollivier–Ricci curvature provided no two faces overlap on multiple edges (quasi-convexity).
• In directed graphs, only feed-forward loop triangles augment the directed Forman–Ricci, fully encoding the orientation structure.

AFRC remains strictly local in computation and effect, in contrast with non-local invariant diffusion-based or optimal-transport-based curvatures. In the absence of non-overlapping short cycles, the augmented formula may overcount or distort, so triangle-only augmentation is typically preferred for general networks (Iváñez, 2022).

7. Limitations and Practical Considerations

While AFRC is highly correlated with Ollivier–Ricci curvature, especially for exploratory and large-scale applications in graphs with abundant small cycles, it does not fully capture global diffusion phenomena intrinsic to metric-measure formalisms. In networks with significant long-range connectivity but poor triangle or quadrangle structure (e.g., trees, certain bipartite graphs), AFRC and Ollivier–Ricci can diverge (Samal et al., 2017, Iváñez, 2022).

In non-quasiconvex networks, augmentation by higher cycles (e.g., 4- or 5-gons) without orientation corrections can lead to distortions; algorithms must explicitly handle possible cycle overlaps or restrict to triangle augmentation to ensure topological fidelity.

AFRC-based workflows rapidly yield curvature distributions supporting scalable graph mining, but final interpretations, especially in applications sensitive to metric properties or requiring metric-consistent curvature notions, should be validated against network-specific ground truth or compared to full Ollivier–Ricci computations.