Graph Ricci Curvature Overview

• Graph Ricci curvature is a family of discrete curvature notions that quantify geometric features like neighborhood overlap, transport contraction, and clustering in graphs.
• Transport-based formulations, including Ollivier and Lin–Lu–Yau models, use optimization over probability measures to capture local network connectivity and bottlenecks.
• Applications span network flow optimization, spectral analysis, and curvature-guided machine learning while linking discrete structures to continuous geometric concepts.

Graph Ricci curvature is a family of discrete curvature notions for graphs that imports the geometric role of Ricci curvature into combinatorial and network settings. In the transport-based formulation of Ollivier, curvature compares the graph distance between two vertices with the 1-Wasserstein distance between probability measures concentrated on their neighborhoods; in the Lin–Lu–Yau formulation, this is refined by a lazy-walk scaling limit as the idleness parameter tends to $1$ (Bhattacharya et al., 2013, Ikeda et al., 2021). Other major formulations include Bakry–Émery curvature derived from $\Gamma/\Gamma_2$ calculus, effective-resistance-based Ricci–Foster curvature, and large-scale or Sobolev-type transport curvatures (Siconolfi, 2021, Dawkins et al., 2024, Kempton et al., 2019, Iwasaki et al., 13 Mar 2026). Across these variants, graph Ricci curvature serves as a quantitative proxy for neighborhood overlap, transport contraction, bottlenecks, spectral gap, diameter control, and diffusion regularity, while remaining sensitive to the chosen metric, Markov kernel, and normalization.

1. Transport-based graph Ricci curvature

In the classical transport-based framework, a graph $G=(V,E)$ is equipped with the shortest-path metric $d$ and a family of vertex measures $m_x$. Ollivier’s coarse Ricci curvature is

$\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$

where $W_1$ is the 1-Wasserstein distance. For unweighted simple graphs with the non-lazy simple random walk,

$$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$

and curvature is evaluated primarily on edges, since an edgewise lower bound propagates to all pairs by Ollivier’s triangle inequality (Bhattacharya et al., 2013). Kantorovich duality gives

$$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$

with $E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$, so curvature can be viewed equivalently as an optimization over 1-Lipschitz potentials (Bhattacharya et al., 2013).

Lin–Lu–Yau curvature modifies Ollivier’s definition by introducing an idleness parameter $\Gamma/\Gamma_2$0 and taking a scaling limit as $\Gamma/\Gamma_2$1. For a weighted graph,

$\Gamma/\Gamma_2$2

and

$\Gamma/\Gamma_2$3

The Lin–Lu–Yau curvature is then

$\Gamma/\Gamma_2$4

and the limit exists (Ikeda et al., 2021). On regular graphs, the idleness function is piecewise linear with at most two linear parts, and for $\Gamma/\Gamma_2$5,

$\Gamma/\Gamma_2$6

which makes the Lin–Lu–Yau quantity especially tractable in the regular setting (Hehl, 2024).

A central interpretation in this transport framework is that curvature measures neighborhood overlap. In the non-lazy setting, common neighbors reduce transport cost and therefore increase curvature. This yields direct links to triangles and local clustering coefficients. For a vertex $\Gamma/\Gamma_2$7, the local clustering coefficient

$\Gamma/\Gamma_2$8

can be written as

$\Gamma/\Gamma_2$9

where $G=(V,E)$0 is the number of common neighbors of $G=(V,E)$1 and $G=(V,E)$2. This relation underlies lower curvature bounds in terms of triangle counts and shows why complete graphs are highly positively curved while trees are extremal on the negative side (Jost et al., 2011).

2. Exact formulas, local structure, and graph classes

A defining feature of transport-based graph Ricci curvature is its local computability. For Ollivier curvature in the non-lazy convention, the curvature of an edge $G=(V,E)$3 depends only on a tightly defined core neighborhood consisting of neighbors of $G=(V,E)$4 and $G=(V,E)$5 and vertices at distance $G=(V,E)$6 from both. The Reduction Lemma states that $G=(V,E)$7 when $G=(V,E)$8 is the induced graph on that core neighborhood, and edges between $G=(V,E)$9 and $d$0 may be removed without changing curvature (Bhattacharya et al., 2013). The associated linear program has a totally unimodular constraint matrix, so optimal potentials may be taken integer-valued and $d$1 is rational (Bhattacharya et al., 2013).

For trees,

$d$2

and the same formula holds on edges free of $d$3-, $d$4-, and $d$5-cycles (Bhattacharya et al., 2013). Triangle-free graphs satisfy

$d$6

while a general upper bound due to Jost–Liu is

$d$7

(Bhattacharya et al., 2013). A matching-based lower bound sharpens this by introducing a maximum matching $d$8 between non-common neighbors: $d$9 which formalizes the idea that matched neighbor mass can be transported at distance $m_x$0 rather than farther away (Bhattacharya et al., 2013).

Several graph classes admit exact or near-exact formulas. For strongly regular graphs with parameters $m_x$1, if $m_x$2 is the size of a maximum matching between the two non-common neighborhood parts $m_x$3 and $m_x$4, then

$m_x$5

where $m_x$6 denotes the condensed Lin–Lu–Yau curvature (Bonini et al., 2019). In girth-$m_x$7 strongly regular graphs this becomes

$m_x$8

so the only such graphs with nonnegative condensed curvature are $m_x$9 and the Petersen graph (Bonini et al., 2019). In girth-$\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$0 strongly regular graphs one gets

$\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$1

because Hall’s theorem produces a perfect matching between the neighborhood shores (Bonini et al., 2019).

Regular graphs admit especially explicit Lin–Lu–Yau formulas. For an edge $\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$2 in a $\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$3-regular graph,

$\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$4

where $\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$5 ranges over bijections from $\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$6 to $\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$7 (Hehl, 2024). This yields a cycle-count form,

$\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$8

which separates contributions from triangles, $\kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)},$9-cycles, and $W_1$0-cycles (Hehl, 2024). The same paper proves that a regular graph has $W_1$1 if and only if it is a cocktail party graph, and that

$W_1$2

(Hehl, 2024).

The gluing construction of two complete graphs illustrates how positive curvature can be engineered combinatorially. If $W_1$3 and $W_1$4 are joined by a connector edge and $W_1$5 additional star edges on each side, then the resulting $W_1$6-gluing graph has positive Ollivier curvature on every edge if and only if

$W_1$7

and the minimum such $W_1$8 is

$W_1$9

The minimum edge curvature is controlled by the cross-star edges and increases with $$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$0 (Yamada, 2018).

3. Alternative graph Ricci curvatures

Bakry–Émery Ricci curvature is vertex-based rather than edge-based and arises from the graph Laplacian through $$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$1-calculus. For the combinatorial Laplacian

$$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$2

one defines

$$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$3

The Bakry–Émery curvature at $$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$4 is the supremum $$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$5 such that

$$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$6

for all $$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$7, equivalently

$$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$8

(Mondal et al., 2024). A related but distinct formulation expresses discrete Ricci curvature as the minimal eigenvalue of a local symmetric matrix $$m_x(y)= \begin{cases} \frac{1}{d_x}, & y\in N(x),\ 0, & \text{otherwise}, \end{cases}$$9 built from the radius-$$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$0 neighborhood; the paper proves

$$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$1

and uses this to compute or bound curvatures of Cayley graphs of finite Coxeter groups and affine Weyl groups (Siconolfi, 2021).

Empirically, Bakry–Émery curvature is typically negative on most vertices in both model and real networks, exhibits a high positive correlation with Ollivier–Ricci and augmented Forman–Ricci curvature, and a high negative correlation with degree and several centrality measures, but does not correlate with the clustering coefficient (Mondal et al., 2024). This suggests that Bakry–Émery curvature captures a broader two-hop cohesion pattern than a purely local triangle ratio.

The paper “Large scale Ricci curvature on graphs” introduces a hybrid between Ollivier and Bakry–Émery curvature by replacing the usual local gradient with an $$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$2-gradient,

$$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$3

and defining curvature notions $$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$4, $$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$5, and $$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$6 through semigroup gradient decay (Kempton et al., 2019). The principal equivalences are: $$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$7

$$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$8

and, on finite graphs,

$$W_1(m_x,m_y)=\sup_{f\text{ 1-Lip}} \{E_x(f)-E_y(f)\},$$9

(Kempton et al., 2019). At scale $E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$0, the hexagonal lattice has nonnegative curvature in this sense (Kempton et al., 2019).

Effective resistance yields yet another notion: Ricci–Foster curvature. For an edge $E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$1 with resistance $E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$2 and effective resistance $E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$3,

$E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$4

This curvature is scale-invariant, satisfies $E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$5 by Foster’s theorem, and drives the Ricci–Foster flow

$E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$6

(Dawkins et al., 2024). The paper proves short-time existence and uniqueness, constant-rate volume decay

$E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$7

and preservation of nonnegative and positive curvature (Dawkins et al., 2024).

Recent work also defines integral Ricci curvature for graphs by measuring the total deficit below a threshold $E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$8. For Lin–Lu–Yau curvature,

$E_x(f):=\sum_{z\in N(x)} f(z)m_x(z)$9

and this leads to Bonnet–Myers-, Moore-, and Lichnerowicz-type estimates depending on $\Gamma/\Gamma_2$00 and $\Gamma/\Gamma_2$01, without requiring the graph to be positively curved everywhere (Olivé, 23 Feb 2025).

4. Higher-order, directed, and hypergraph generalizations

Graph Ricci curvature has been extended to several higher-order discrete structures. On simplicial complexes, the paper “The Ricci curvature on simplicial complexes” defines probability measures on $\Gamma/\Gamma_2$02-faces through shared $\Gamma/\Gamma_2$03-cofaces and sets

$\Gamma/\Gamma_2$04

For adjacent $\Gamma/\Gamma_2$05-faces $\Gamma/\Gamma_2$06, the paper proves explicit upper and lower bounds depending on $\Gamma/\Gamma_2$07, $\Gamma/\Gamma_2$08, and $\Gamma/\Gamma_2$09, and derives eigenvalue estimates for the normalized up-Laplacian: $\Gamma/\Gamma_2$10 under the curvature lower bound $\Gamma/\Gamma_2$11 (Yamada, 2019).

Directed graphs require a nonsymmetric metric and out-neighborhood measures. For a strongly connected directed graph and $\Gamma/\Gamma_2$12,

$\Gamma/\Gamma_2$13

$\Gamma/\Gamma_2$14

The paper proves upper and lower bounds, criteria for Ricci-flat directed regular graphs, and curvature identities for Cartesian products of directed graphs (Yamada, 2016).

Hypergraphs admit multiple non-equivalent generalizations. One approach uses a nonlinear submodular Laplacian. For a weighted finite connected hypergraph $\Gamma/\Gamma_2$15, the normalized Laplacian is

$\Gamma/\Gamma_2$16

where $\Gamma/\Gamma_2$17 is the multivalued submodular hypergraph Laplacian and the resolvent

$\Gamma/\Gamma_2$18

is single-valued, continuous, and non-expansive (Ikeda et al., 2021). Using a weighted Lipschitz class,

$\Gamma/\Gamma_2$19

and the infinitesimal limit

$\Gamma/\Gamma_2$20

exists on hypergraphs and reduces exactly to Lin–Lu–Yau curvature on graphs (Ikeda et al., 2021). This curvature yields a spectral gap bound

$\Gamma/\Gamma_2$21

a gradient estimate for heat flow,

$\Gamma/\Gamma_2$22

and a Bonnet–Myers-type diameter bound

$\Gamma/\Gamma_2$23

under positive upper curvature (Ikeda et al., 2021).

A distinct hypergraph notion is Hypergraph Lower Ricci Curvature (HLRC), a closed-form curvature defined on hyperedges: $\Gamma/\Gamma_2$24 It satisfies

$\Gamma/\Gamma_2$25

for $\Gamma/\Gamma_2$26, reduces in the 2-uniform case to the graph lower Ricci curvature formula of Park–Li, and is computationally much cheaper than hypergraph Ollivier–Ricci curvature (Yang et al., 4 Jun 2025).

5. Products, asymptotics, and continuum limits

Product constructions reveal how discrete curvature behaves under graph composition. For the Cartesian product $\Gamma/\Gamma_2$27 of regular graphs,

$\Gamma/\Gamma_2$28

$\Gamma/\Gamma_2$29

with symmetric formulas for vertical edges, where $\Gamma/\Gamma_2$30 (Mou, 16 Jun 2025). The same paper gives explicit formulas for horizontal and vertical edges of the strong product $\Gamma/\Gamma_2$31: $\Gamma/\Gamma_2$32

$\Gamma/\Gamma_2$33

with symmetric formulas for vertical edges, but proves by counterexample that no comparably simple general formula exists for diagonal edges (Mou, 16 Jun 2025).

Random graph asymptotics exhibit phase transitions in curvature. For $\Gamma/\Gamma_2$34, conditioned on a fixed edge $\Gamma/\Gamma_2$35, the non-lazy Ollivier curvature satisfies:

• if $\Gamma/\Gamma_2$36, then $\Gamma/\Gamma_2$37;
• if $\Gamma/\Gamma_2$38, then

$\Gamma/\Gamma_2$39

with $\Gamma/\Gamma_2$40 i.i.d. Poisson($\Gamma/\Gamma_2$41);

• if $\Gamma/\Gamma_2$42 and $\Gamma/\Gamma_2$43, then $\Gamma/\Gamma_2$44;
• if $\Gamma/\Gamma_2$45 and $\Gamma/\Gamma_2$46, then $\Gamma/\Gamma_2$47;
• if $\Gamma/\Gamma_2$48 and $\Gamma/\Gamma_2$49, then $\Gamma/\Gamma_2$50;
• if $\Gamma/\Gamma_2$51, then $\Gamma/\Gamma_2$52 (Bhattacharya et al., 2013). Analogous regimes hold for random bipartite graphs $\Gamma/\Gamma_2$53 (Bhattacharya et al., 2013).

A different asymptotic direction connects discrete and smooth geometry. For random geometric graphs on a compact Riemannian manifold, equipped with either a manifold-weighted shortest-path metric or a rescaled hop metric, the paper proves that Ollivier–Ricci curvature converges to the manifold Ricci curvature: $\Gamma/\Gamma_2$54 under explicit scaling regimes for the connection radius $\Gamma/\Gamma_2$55 and measure radius $\Gamma/\Gamma_2$56 (Hoorn et al., 2020). This is described there as the first rigorous result linking curvature of random graphs to the Ricci curvature of the underlying manifold (Hoorn et al., 2020).

6. Flows, network applications, and current directions

Ricci curvature has become a practical network observable because negative curvature detects bottlenecks and positive curvature detects cohesive substructures. In “Core detection via Ricci curvature flows on weighted graphs,” several discrete curvature flows are studied on weighted graphs. One representative update is

$\Gamma/\Gamma_2$57

where $\Gamma/\Gamma_2$58 is the shortest-path distance between the endpoints of edge $\Gamma/\Gamma_2$59 under the current weights (Zhao et al., 2 Aug 2025). The paper derives explicit upper and lower bounds on edge weights along such flows and uses them to prove that, for suitable step sizes and a bounded number of iterations, weights neither overflow nor become numerically zero (Zhao et al., 2 Aug 2025). It then applies Ricci flow to core-subgraph detection and reports better performance than PageRank, degree, betweenness, and closeness centrality on Cora, Citeseer, and Bio-CE-HT (Zhao et al., 2 Aug 2025).

Resistance-based curvature also supports a flow theory. The Ricci–Foster flow

$\Gamma/\Gamma_2$60

decreases the total length at unit rate, preserves nonnegative curvature, and admits surgery when an edge length reaches zero (Dawkins et al., 2024). In this framework, cycles shrink homothetically and play the role of discrete Einstein metrics, while trees exhibit mixed shrinking and expansion behavior depending on vertex degrees (Dawkins et al., 2024).

Curvature has also become relevant to graph machine learning. In the context of GNN-based SAT solving, the paper “On the Hardness of Learning GNN-based SAT Solvers: The Role of Graph Ricci Curvature” uses Balanced Forman Curvature as its principal curvature notion and Ollivier–Ricci lower bounds as a comparison tool (Skenderi, 29 Aug 2025). For literal–clause bipartite graphs of random $\Gamma/\Gamma_2$61-SAT formulas, it proves that the average curvature becomes increasingly negative as clause density $\Gamma/\Gamma_2$62 grows and, asymptotically,

$\Gamma/\Gamma_2$63

which is strictly negative for $\Gamma/\Gamma_2$64 (Skenderi, 29 Aug 2025). The paper connects this to oversquashing in GNNs and reports that curvature-guided rewiring significantly improves solver accuracy (Skenderi, 29 Aug 2025). A plausible implication is that Ricci curvature can serve not only as a structural diagnostic but also as a proxy for message-passing hardness when long-range dependencies are forced through negatively curved edges.

Another recent transport-based proposal is Sobolev–Ricci Curvature (SRC), defined from a tree-metric Sobolev transport distance

$\Gamma/\Gamma_2$65

with

$\Gamma/\Gamma_2$66

SRC recovers Ollivier–Ricci curvature on trees with the length measure when $\Gamma/\Gamma_2$67, vanishes in the Dirac limit, and is used in Sobolev–Ricci flow and curvature-guided edge pruning (Iwasaki et al., 13 Mar 2026). This suggests a current trend toward transport-based curvatures that preserve geometric meaning while reducing the computational burden of optimal transport.

A recurrent misconception is that “graph Ricci curvature” denotes a single invariant. The literature surveyed here shows instead that it is a family of inequivalent constructions—transport-based, Laplacian-based, resistance-based, combinatorial, and higher-order—each emphasizing different local or mesoscopic features. Another common misconception is that exact formulas are broadly available. In fact, exact formulas are confined to special classes such as trees, regular graphs, strongly regular graphs, or selected graph products, and even there caveats remain: an earlier claim of exact formulas for some bipartite and girth-$\Gamma/\Gamma_2$68 classes was later invalidated in part by a bug, leaving upper bounds intact but exactness open (Bhattacharya et al., 2013). The current state of the subject is therefore both structurally rich and technically heterogeneous: discrete Ricci curvature is now tied to spectral theory, diameter bounds, clustering, random graphs, manifold limits, curvature flows, higher-order networks, and machine-learning pathologies, but its precise meaning always depends on the chosen definition and the graph model under study.