Forman-Ricci Curvature in Discrete Structures

Updated 17 November 2025

Forman-Ricci curvature is a discrete combinatorial measure that extends Ricci curvature to graphs, hypergraphs, and higher-dimensional structures.
It assesses local edge properties by analyzing counts, weights, and incidence relations, enabling insights into network connectivity and topology.
Recent developments extend its application to directed networks, hypergraphs, and persistent homology, enhancing scalable analysis of complex systems.

Forman-Ricci curvature is a discrete, combinatorial measure of curvature for cell complexes—foremost graphs, hypergraphs, and their higher-dimensional generalizations—defined by Robin Forman as an analogue of Ricci curvature for smooth Riemannian manifolds. In the graph-theoretic setting, Forman-Ricci curvature is an edge-centric scalar that quantifies the local divergence or cohesion of flows around each edge, using nothing but counts and weights of adjacent cells. Since its introduction, it has become a powerful and scalable tool for probing network connectivity, robustness, organization, community structure, information dynamics, and topological features. Recent advances further extend Forman-Ricci curvature to directed networks, weighted complexes, hypergraphs, higher-dimensional simplicial structures, and persistent homology filtrations.

1. Foundational Definitions and Formulas

Forman-Ricci curvature on a CW-complex (or graph, simplicial complex, or general cell complex) considers each $p$ -cell $\alpha$ and accounts for its local incidence relations. The central formula (Bloch, 2014, Sreejith et al., 2016, Sreejith et al., 2016, Samal et al., 2017) is

$\ricci(\alpha) = \#\{\text{%%%%2%%%%–cells } \beta > \alpha\} + \#\{\text{%%%%3%%%%–cells } \gamma < \alpha\} - \#\{\text{parallel %%%%4%%%%–cells}\},$

where:

$\beta > \alpha$ means $\alpha$ is a face of $\beta$ (coface);
$\gamma < \alpha$ means $\gamma$ is a face of $\alpha$ ;
"parallel $p$ –cells" are distinct $p$ -cells sharing either a common $(p+1)$ –coface or a common $(p-1)$ -face, but not both.

For edges in a graph ( $p=1$ ), with default combinatorial weights,

$\ricci(e) = 4 - \deg(v_1) - \deg(v_2)$

for an edge $e = (v_1, v_2)$ in an undirected, unweighted graph (Sreejith et al., 2016, Sreejith et al., 2016, Sreejith et al., 2016). If weights $w_e>0$ , $w_{v_1}>0$ , $w_{v_2}>0$ are available (possibly nontrivial), substitute: $\mathbf{F}(e) = w_e \left( \frac{w_{v_1}}{w_e} + \frac{w_{v_2}}{w_e} - \sum_{e' \sim v_1,\, e' \ne e} \frac{w_{v_1}}{\sqrt{w_e w_{e'}}} - \sum_{e' \sim v_2,\, e' \ne e} \frac{w_{v_2}}{\sqrt{w_e w_{e'}}} \right)$ (Sreejith et al., 2016, Samal et al., 2017).

Directed graphs require adapting the neighbor sets according to edge orientation: $F(e = v_1 \to v_2) = w_e \left( \frac{w_{v_1}}{w_e} - \sum_{e'\in \mathrm{In}(v_1) \setminus \{e\}} \frac{w_{v_1}}{\sqrt{w_e w_{e'}}} \right) + w_e \left( \frac{w_{v_2}}{w_e} - \sum_{e''\in \mathrm{Out}(v_2) \setminus \{e\}} \frac{w_{v_2}}{\sqrt{w_e w_{e''}}} \right)$ (Sreejith et al., 2016, Saucan et al., 2018, Pouryahya et al., 2017).

Node-level (scalar) curvature is obtained by summing or averaging incident edge curvatures: $\mathbf{F}_{\mathrm{unn}}(v) = \sum_{e \sim v} \mathbf{F}(e), \qquad \mathbf{F}_{\mathrm{norm}}(v) = \frac{1}{\deg(v)} \sum_{e \sim v} \mathbf{F}(e)$ (Sreejith et al., 2016, Sreejith et al., 2016).

2. Geometric and Network-Theoretic Interpretation

Forman-Ricci curvature ascribes geometric intuition from smooth Ricci curvature to discrete contexts. High positive curvature ( $>0$ ) at an edge signals locally “convergent” or tree-like structure (leaves, clique cores). Zero curvature marks “flat” regular graphs (lattices, cycles, grid patches). Strongly negative curvature indicates “divergent,” hyperbolic, or bottlenecked regions (edges connecting hubs, community bridges) (Sreejith et al., 2016, Wayland et al., 27 Aug 2024, Beuria, 2023).

In networks:

Negative curvature localizes bottlenecks vulnerable to loss of global connectivity (removal quickly fragments network).
Positive curvature occurs in redundant or highly clustered regions.
Distributional properties distinguish random, small-world, and scale-free architectures (Sreejith et al., 2016).
Curvature is strongly negatively correlated with degree and centrality measures (betweenness, closeness); node or edge removal by curvature disrupts communication nearly as efficiently as removals by high centrality (Sreejith et al., 2016, Sreejith et al., 2016).

Recent work leverages curvature in dynamic settings:

Ricci flow on edge weights ( $w_e$ ) to denoise or contract/expand regions adaptively (Weber et al., 2016).
Persistent homology filtration by curvature—edges/vertices appear in order dictated by curvature—producing topologically meaningful barcodes (Iváñez, 2022, Saucan, 2020, Roy et al., 2019, Beuria, 2023).

3. High-Order, Augmented, and Hypergraph Curvature

Forman’s approach extends naturally to higher-order structures:

For a $d$ -simplex $\alpha$ , the formula generalizes to

$F_d(\alpha) = |H^d_\alpha| + (d+1) - |P^d_\alpha|$

where $H^d_\alpha$ is the set of cofaces (dimension $d+1$ ), $P^d_\alpha$ the parallel neighbors (Souza et al., 2023, Souza et al., 30 Apr 2025).

Efficient computation exploits set-intersections of node neighborhoods; local update rules scale to large datasets (“FastForman” algorithm) (Souza et al., 2023, Souza et al., 30 Apr 2025).

Augmented Forman-Ricci curvature (AFRC) accounts for higher-order cycles—e.g., triangles, squares—by treating cycles as 2-cells: $\mathcal{AF}_3(u,v) = 4 - \deg(u) - \deg(v) + 3 \cdot \triangle(u,v)$ where $\triangle(u,v)$ is the number of triangles containing edge $(u,v)$ (Fesser et al., 2023, Fesser et al., 2023, Iváñez, 2022). In non-quasiconvex complexes, general augmentation must track face overlaps and orientation, complicating computation (Iváñez, 2022).

For hypergraphs, Forman-Ricci curvature quantifies the trade-off between hyperedge size and degree: $F(e) = 2|e| - \sum_{k \in e} d_k$ with $d_k$ the hypergraph degree of $k$ (Leal et al., 2018, Saucan et al., 2018). Directed hyperarcs admit four elementary curvatures measuring flow redundancy and bottlenecks (Leal et al., 2018). These can be decomposed according to in/out degrees of source and sink sets.

4. Algorithmic Implementation and Computational Considerations

Forman-Ricci curvature is fundamentally local: for graphs, the cost per edge is $O(\deg(v_1) + \deg(v_2))$ , and linear in edge count for bounded-degree graphs (Sreejith et al., 2016, Sreejith et al., 2016, Banf et al., 15 Aug 2025). Node and edge weights, as well as directed structure, are handled in-place with array/dictionary lookups.

Higher-dimensional cases entail clique finding; set-intersection acceleration reduces overhead (Souza et al., 2023, Souza et al., 30 Apr 2025). Persistent homology filtrations by curvature require sorting simplex birth times, pushing computational cost to the enumeration of complex cliques (NP-hard in general).

Curvature computation parallelizes trivially; each edge's curvature is independent, and final node-accumulation is linear (Sreejith et al., 2016, Souza et al., 2023). Hypergraph curvature admits similar locality if hyperedges are stored as vertex lists. Practical use cases scale to graphs with millions of edges (Wayland et al., 27 Aug 2024).

Curvature-augmented rewiring algorithms (GNN over-squashing mitigation) rely on rapid curvature computation and empirical curvature distribution thresholds; Gaussian mixture fits segment bridge vs. cluster edges for edge addition/removal (Fesser et al., 2023, Banf et al., 15 Aug 2025). AFR-based rewiring achieves order-of-magnitude speed-up over Ollivier-Ricci methods.

5. Extensions: Directed, Signed-Control, and Weighted Networks

Forman-Ricci curvature adapts naturally to directed graphs:

At each edge $e = v_1 \to v_2$ , sum over In $(v_1)$ and Out $(v_2)$ , yielding:

$F(e) = w_e \left( \frac{w_{v_1}}{w_e} - \sum_{e' \in \mathrm{In}(v_1) \setminus \{e\}} \frac{w_{v_1}}{\sqrt{w_e w_{e'}}} \right) + w_e \left( \frac{w_{v_2}}{w_e} - \sum_{e'' \in \mathrm{Out}(v_2) \setminus \{e\}} \frac{w_{v_2}}{\sqrt{w_e w_{e''}}} \right)$

(Sreejith et al., 2016, Saucan et al., 2018, Pouryahya et al., 2017).

In biological networks, “signed-control” Ricci curvature introduces +1 or –1 multipliers to parallel neighbor contributions according to activator/repressor edge labels (Pouryahya et al., 2017).

Weighted networks use node/edge weights directly in the base formula. Default combinatorial weights $(= 1)$ generally suffice, but domain-specific weights (length, traffic, mass) refine geometric sensitivity (Saucan et al., 2018, Beuria, 2023, Weber et al., 2016).

6. Applications and Comparative Perspectives

Forman-Ricci curvature has demonstrated utility in:

Network classification, bottleneck identification (Sreejith et al., 2016, Sreejith et al., 2016, Banf et al., 15 Aug 2025, Wayland et al., 27 Aug 2024).
Vulnerability analysis, attack strategies (Sreejith et al., 2016).
Community detection—AFRC-based edge deletion rivals Ollivier-Ricci curvature methods in accuracy, at much lower computational cost (Fesser et al., 2023, Iváñez, 2022).
Over-squashing and over-smoothing diagnosis and mitigation in GNNs, by curvature-driven rewiring and structural lifting (Banf et al., 15 Aug 2025, Fesser et al., 2023).
Persistent homology and topological data analysis, curvature-induced filtrations for barcode extraction (Saucan, 2020, Roy et al., 2019, Iváñez, 2022, Souza et al., 30 Apr 2025, Beuria, 2023).
Medical informatics, social/bio networks, transportation, and physics, via large-scale, interpretable descriptions of system structure (Wayland et al., 27 Aug 2024, Beuria, 2023).

Forman-Ricci curvature correlates strongly with classical centralities (negatively) but weakly with local clustering; negatively curved edges are critical for connectivity, but clustering coefficient does not in general predict curvature (Sreejith et al., 2016, Sreejith et al., 2016, Samal et al., 2017, Pouryahya et al., 2017).

Comparisons with Ollivier-Ricci curvature across model and real networks reveal strong empirical correlation (Samal et al., 2017, Fesser et al., 2023, Wayland et al., 27 Aug 2024), especially with augmented (triangle-aware) Forman curvature. Ollivier-Ricci, grounded in optimal transport, offers richer metric-theoretic interpretation but is far more expensive to compute.

7. Topological, Homological, and Future Directions

The extension of Forman-Ricci curvature to persistent homology is well-justified: filtration by curvature closely tracks discrete Morse filtration and recovers combinatorial analogues of Gauss–Bonnet (Bloch, 2014, Saucan, 2020, Iváñez, 2022). Explicit discrete Gauss–Bonnet formulas for vertices, edges, and faces restore topological invariants lost in naïve edge-only curvature (Bloch, 2014).

Current research continues to develop efficient algorithms for high-order curvature, non-quasiconvex complexes, and general CW structures (Souza et al., 2023, Souza et al., 30 Apr 2025, Iváñez, 2022); investigates signed and directed extensions for biological systems (Pouryahya et al., 2017); adapts curvature as a filter for geometry-aware machine learning pipelines (Souza et al., 30 Apr 2025); and benchmarks alternative discretizations.

Open questions include full integration of orientation and signed covariants for generalized augmentation, extension to arbitrary weighted, directed hypergraphs, and theoretical characterization of curvature-induced topological transitions in large networks.

Forman-Ricci curvature thus provides a scalable, robust, and flexible geometric descriptor for discrete structures, combining combinatorial simplicity with significant topological and analytical expressivity. It serves as both an analytic tool for network science and a geometric foundation for computational topology and graph-based learning.