Forman Ricci Curvature: A Discrete Geometric Approach

Updated 29 November 2025

Forman Ricci curvature is a combinatorial discretization that assigns meaningful curvature measures to cells in networks and complexes.
It encodes local connectivity and geometric properties to identify bottlenecks, community structures, and network robustness.
Efficient algorithms leverage edge, triangle, and higher-order computations to support topological data analysis and geometric deep learning.

Forman Ricci curvature is a combinatorial discretization of the Ricci curvature concept from Riemannian geometry, designed to assign a meaningful "curvature" value to each cell (vertex, edge, higher simplex) of a finite cell complex, most notably graphs, simplicial complexes, or more general CW complexes. It encodes geometric information about local connectivity and network architecture, is computationally efficient, and extends naturally to higher-order structures such as hypergraphs and cell complexes. Since its introduction, Forman Ricci curvature has seen broad adoption in applied mathematics, data science, complex networks, geometric deep learning, and computational topology.

1. Mathematical Definition and Generalization

Forman Ricci curvature originates from a discrete analogue of Bochner–Weitzenböck theory on cell complexes. Given a regular CW complex $K$ or, more generally, a (possibly weighted) abstract cell complex, Forman’s curvature is defined for each $p$ -cell %%%%2%%%% by

$\operatorname{Ricci}(\alpha) = |\{\beta > \alpha\}| + |\{\gamma < \alpha\}| - |\{\text{parallel neighbors of } \alpha\}|$

where:

$\{\beta > \alpha\}$ denotes the set of $(p+1)$ -cells with $\alpha$ as a face,
$\{\gamma < \alpha\}$ denotes the set of $(p-1)$ -faces of $\alpha$ ,
Parallel neighbors are other $p$ -cells sharing exactly a $(p+1)$ -coface or a $(p-1)$ -subface with $\alpha$ , but not both (Bloch, 2014).

In the setting of graphs (CW complexes of dimension 1), this reduces to the canonical (possibly weighted) edge formula: $F(e) = w_e \left( \frac{w_{v_1}}{w_e} + \frac{w_{v_2}}{w_e} - \sum_{e'\sim v_1, e'\neq e} \frac{w_{v_1}}{\sqrt{w_e w_{e'}}} - \sum_{e'\sim v_2, e'\neq e} \frac{w_{v_2}}{\sqrt{w_e w_{e'}}} \right)$ where $w_e$ is the weight of edge $e$ , $w_{v_i}$ the weights of its endpoints, and "parallel" edges are those sharing one endpoint but not forming higher faces (Sreejith et al., 2016, Weber et al., 2016, Saucan, 2019).

Unweighted, undirected graph: If all weights are 1, this collapses to

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

This basic version is highly scalable and forms the foundation for discrete network curvature analysis (Sreejith et al., 2016).

Weighted/directed/complexes: Extensions incorporate arbitrary cell weights, edge directions, and higher-dimensional faces (for simplicial complexes, hypergraphs, or general CW-complexes) (Leal et al., 2018, Saucan et al., 2018, Sreejith et al., 2016). For hypergraphs, the curvature of a hyperedge encodes a trade-off between its cardinality and the sum of the participation degrees of its constituent vertices (Leal et al., 2018).

For higher-dimensional simplices (e.g., triangles, tetrahedra), the curvature of a $d$ -cell $\alpha$ takes the form

$F_d(\alpha) = |\{\beta \in C_{d+1} : \beta > \alpha\}| + (d+1) - |\{\alpha' \,||\, \alpha\}|$

where parallel neighbors $\alpha' \,||\, \alpha$ are as defined above (Souza et al., 2023, Souza et al., 30 Apr 2025).

2. Augmented and Higher-Order Curvature

Original Forman Ricci curvature on graphs is a degree-based, essentially 1-local invariant. However, the inclusion of higher-order structures (cycles, triangles, cliques) substantially increases discriminatory power in complex networks.

Augmented Forman Ricci Curvature (AFRC): Given a CW-complex where short cycles (e.g., triangles) are "filled in" as 2-cells, the curvature of an edge e acquires positive contributions from each incident triangle: $AF(e) = 4 - \deg(v_1) - \deg(v_2) + 3 \times \text{number of triangles containing } e$ This idea generalizes to cycles of any length, with coefficients decreasing for longer cycles: each triangle supplies +3, each 4-cycle +2, each 5-cycle +1, and so on (Samal et al., 2017, Iváñez, 2022).

For networks with significant higher-order architecture, AFRC much more accurately tracks network robustness, bottlenecks, and community structure. AFRC is computationally efficient: triangle counts can be performed with local neighbor list intersections, and further augmentations (squares, pentagons) are tractable for moderate graph sizes (Fesser et al., 2023, Souza et al., 2023, Iváñez, 2022).

Connection to Ollivier–Ricci curvature: Despite differing mathematical underpinnings—Forman is combinatorial, Ollivier is optimal-transport—the two are often highly correlated on empirical networks, especially when AFRC includes higher-order terms (Samal et al., 2017, Fesser et al., 2023). This suggests AFRC as a practical surrogate for Ollivier–Ricci curvature in computationally-constrained scenarios.

3. Computational Methods and Complexity

Edge-centric algorithm (F(e) - basic):

for each edge e = (u, v):
    A = sum_{e' ≠ e, e' incident to u} (w(u) / sqrt(w(e) * w(e')))
    B = sum_{e' ≠ e, e' incident to v} (w(v) / sqrt(w(e) * w(e')))
    F(e) = w(e) * (w(u) / w(e) + w(v) / w(e) - A - B)

For unweighted graphs:

F(e) = 4 - \deg(u) - \deg(v)

(Sreejith et al., 2016, Weber et al., 2016).

Augmented (triangle-aware) version:

1
2
3

for each edge (u, v):
    t = |neighbors(u) ∩ neighbors(v)|    # triangle count
    F_aug(e) = 4 - deg(u) - deg(v) + 3*t

Both are linear in the number of edges for sparse graphs (Sreejith et al., 2016, Fesser et al., 2023).

Higher-dimensional (simplicial complex) FastForman algorithm: Uses set-theoretic representations of neighborhoods and intersection operations to avoid expensive enumeration of faces/cofaces or parallel neighbors (Souza et al., 2023, Souza et al., 30 Apr 2025).

Directed graphs: Edge orientation restricts neighbor sums to appropriate sources/incomings and targets/outgoings: $F(e = u \to v) = 2 - \deg^-(u) - \deg^+(v) \quad (\text{unweighted})$ Weighted versions adjust the sums accordingly, and extensions for signed-control curvature allow for edges with positive/negative control (e.g., regulatory or repressive links) (Sreejith et al., 2016, Pouryahya et al., 2017).

Hypergraphs and polyhedral complexes: Analogous expressions depend on both hyperedge size and vertices' hyperedge-participation degrees; the curvature quantifies a trade-off between size (creation) and sharedness (dispersion) (Leal et al., 2018, Saucan et al., 2018).

4. Theoretical Properties and Structural Interpretation

Locality: Forman Ricci curvature is solely a function of local adjacency relations and, in augmented versions, local cycles. No global optimization is involved.
Significance of values: Strongly negative curvature flags bottlenecks, bridges, or hub edges. Large positive curvature indicates edges lying in tightly connected, redundant clusters (e.g., cliques).
Topology and geometry: In higher dimensions, the theory recovers Gauss–Bonnet–type theorems for finite ranked posets and polyhedral complexes (Bloch, 2014). For rank-2 (cell) complexes, the sum of appropriately defined vertex, edge, and face curvatures recovers the Euler characteristic.
Robustness: Curvature provides a proxy for network robustness: edges of high positive curvature contribute to local coherence, whereas negative curvature edges are associated with fragility or vulnerability. This interpretation is supported by attack-vulnerability and percolation studies (Sreejith et al., 2016, Pouryahya et al., 2017).
Correlation with classical measures: FRC is significantly (negatively) correlated with node degree, betweenness, closeness, and eigenvector centrality; less so with clustering coefficient. However, augmentation with triangles increases sensitivity to local clustering (Sreejith et al., 2016, Pouryahya et al., 2017).

Comparative summary:

Property	Forman Ricci	Augmented Forman	Ollivier Ricci
Computational cost	Linear in	E
Local vs global	Purely local	Local + small cycles	Local transport
Captures cycles/triangles	No	Yes	Yes
Sensitivity to bottlenecks/hubs	High	High	High
Theoretical link to geometry	Combinatorial	Via higher-order cells	Optimal transport

5. Extensions: Directed, Weighted, Hypergraphs, and Higher Complexes

Directed networks: Forman curvature extends to directed graphs by orienting the "neighbor" sums: incoming arcs sum at the source, outgoing arcs at the target. Signed-control curvature incorporates positive/negative regulatory logic in biological networks, controlling the sign of curvature contributions (Pouryahya et al., 2017, Sreejith et al., 2016).
Weighted networks: Any positive weighting of vertices, edges, faces can be accommodated, with amplification or attenuation of curvature accordingly. In undirected graphs, node-weights may be set as degree means; edges may be weighted by length, similarity, or data-derived quantities (Weber et al., 2016, Saucan, 2019).
Hypergraphs and polyhedral complexes: For an undirected (hyper)edge $e$ containing vertices $k$ ,

$F(e) = 2|e| - \sum_{k \in e} d_k$

where $d_k$ is the degree (hyperedge participation) of $k$ (Leal et al., 2018). The curvature thus distinguishes between large, exclusive hyperedges (positive curvature) and highly shared, "bridge" hyperedges (negative curvature).

Higher-dimensional complexes/simplices: Formulae exist for arbitrary $d$ -simplices, with "parallel neighbor" combinatorics determined by face and coface relationships (Souza et al., 2023, Souza et al., 30 Apr 2025).

6. Applications and Empirical Results

Network science: FRC distinguishes random, small-world, and scale-free models by the distribution and range of curvature values (Sreejith et al., 2016). It predicts network vulnerability: targeted removal of highly negative curvature nodes/edges fragments networks more efficiently than random deletion (Sreejith et al., 2016, Pouryahya et al., 2017).
Community and anomaly detection: AFRC is directly competitive with (and much faster than) Ollivier–Ricci curvature in algorithmic tasks such as community detection, identification of bridges/bottlenecks, and network denoising (Fesser et al., 2023, Samal et al., 2017, Weber et al., 2016).
Geometric and topological data analysis: FRC-based filtrations support persistent homology and higher-order topological summaries, with barcodes providing network "fingerprints" for classification and anomaly detection (Iváñez, 2022, Saucan, 2020, Roy et al., 2019).
Machine learning on graphs: Curvature features, both at the edge and node level, are used as input to manifold learning and clustering, as well as in the mitigation of over-squashing and over-smoothing in GNN architectures. Forman curvature–based rewiring substantially improves performance and scalability relative to spectral and Ollivier–Ricci approaches (Banf et al., 15 Aug 2025, Fesser et al., 2023, Souza et al., 30 Apr 2025).
Hypergraph analysis: In biological, social, and metabolic systems modeled as hypergraphs, Forman curvature identifies central bottlenecks and redundant structures (Leal et al., 2018, Saucan et al., 2018).

7. Connections to Persistent Homology and Discrete Morse Theory

The use of Forman Ricci curvature as a Morse function or filtration parameter in persistent homology exploits its locality and sensitivity to topology (Iváñez, 2022, Roy et al., 2019, Saucan, 2020). Standard filtrations derived from curvature order edges and simplices by geometric "importance," enabling topological multi-scale analyses that are robust and interpretable. Theoretical results connect curvature filtration to classical combinatorial and Morse-theoretic invariants (vertex defect, critical points) (Saucan, 2020).

8. Limitations and Open Problems

Dependence on local structure: FRC is insensitive to long-range global geometry unless augmented.
Bipartite and sparse graphs: Unaugmented FRC fails to detect community structure in tree-like or bipartite cases; additional cycle-based augmentation is sometimes needed (Iváñez, 2022, Fesser et al., 2023).
Non-quasiconvexity: In non-quasiconvex complexes, extended AFRC requires orientation-aware corrections to avoid multiplicity/overcounting; new formulae are under development (Iváñez, 2022).
Scalability: While basic FRC is linear in network size, the computation of large numbers of higher-order cycles or hyperedges remains costly unless bounded to triangles/quadrangles or restricted to moderate graph density (Souza et al., 2023, Fesser et al., 2023).
Generalization to signed or negative weights: Extending FRC to signed graphs or networks with negative (e.g., inhibitory or competitive) weights remains an open research challenge (Sreejith et al., 2016, Pouryahya et al., 2017).

Forman Ricci curvature provides an edge- or cell-centric, local, combinatorial characterization of discrete geometries. Its theoretical foundations ensure compatibility with curvature-driven flows, Gauss–Bonnet analogues, and Morse-theoretic filtrations, while augmentations enable sensitivity to higher-order structures. Due to its computational tractability, it is a preferred tool for large-scale network analysis, machine learning applications, and topological data analysis tasks where geometric signal and interpretability are both crucial.