Discrete Ricci Flow Algorithms

• Discrete Ricci flow algorithms are a family of computational methods that evolve finite-dimensional metric variables using curvature-driven updates on discretized geometric structures.
• These methods are implemented in various settings—including Kähler geometry, triangulated surfaces, piecewise-flat manifolds, and weighted graphs—each with its own update schemes and convergence criteria.
• Convergence is achieved through strategies such as convex energy minimization, implicit schemes, or surgical interventions, ensuring the transition to constant or balanced curvature states.

A discrete Ricci flow algorithm is a discrete analogue of Ricci flow in which the evolving object is not a smooth Riemannian metric tensor on a continuum manifold, but a finite set of geometric variables such as Kähler potentials in a fixed class, circle-packing or discrete conformal factors on a triangulated surface, edge lengths on a simplicial or piecewise-flat complex, or edge weights on a weighted graph. In current usage, the term therefore denotes a family of algorithms rather than a single construction. Representative instances include the Ricci iteration toward Kähler–Einstein or cscK metrics, unified surface Ricci flow on triangular meshes, Regge–Ricci and piecewise-flat flows in three dimensions, and Ollivier- or LLY-based flows on weighted graphs (Darvas et al., 2017, Zhang, 2023, zhang et al., 2014, Miller et al., 2013, Bai et al., 2020).

1. Scope, variables, and discrete geometric data

The defining feature across these algorithms is that curvature is represented by finite-dimensional data and used to update finite-dimensional metric variables. On compact Kähler manifolds, one works in the space of smooth Kähler potentials

$$\mathcal H := \{\varphi \in C^\infty(M): \omega_\varphi := \omega_0 + i\partial\bar\partial \varphi > 0\},$$

with Ricci form, scalar curvature, and Mabuchi-type functionals driving an implicit discrete evolution (Zhang, 2023). On triangulated surfaces, the state variable is typically a discrete conformal factor $u_i$ at each vertex, with edge lengths reconstructed from $u_i$, background geometry, and scheme parameters $(\varepsilon,\eta)$ (zhang et al., 2014). On piecewise-flat manifolds, the state variables are primal edge lengths $\{\ell_i\}$, and curvature is concentrated on codimension-$2$ hinges through deficit angles (Miller et al., 2013, Conboye et al., 2016). On graphs, the state is a positive edge-weight function $w:E\to(0,\infty)$, and curvature is computed edgewise from optimal transport or related discrete curvature notions (Ni et al., 2018, Ma et al., 21 May 2025).

The update rule depends on the setting. In graph formulations inspired by Ollivier, the continuous prototype is

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$

or its discrete version

$$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$

possibly followed by normalization (Hehl et al., 26 Sep 2025). In Kähler geometry, the discrete step is typically an implicit Euler scheme for a geometric flow, so the new metric appears inside the Ricci term itself (Zhang, 2023, Darvas et al., 2017). In simplicial Regge-type constructions, the update is an ODE for edge lengths or dual-edge lengths expressed through discrete Ricci tensors averaged over hybrid volumes (Miller et al., 2013). This diversity is fundamental: the phrase “discrete Ricci flow” does not identify a unique discretization, a unique curvature, or a unique target geometry.

A recurrent misconception is to treat graph Ricci flow, simplicial Regge–Ricci flow, and Kähler Ricci iteration as interchangeable. They are not. Graph algorithms evolve weighted shortest-path geometries and neighborhood transport data; simplicial algorithms evolve piecewise-flat metrics through deficit angles and dual cells; Kähler algorithms evolve metrics inside a fixed Kähler class through complex Monge–Ampère equations. The common thread is curvature-driven metric deformation, not a shared numerical representation.

2. Ricci iteration in Kähler geometry

In the Kähler setting, a discrete Ricci flow algorithm is often a backward Euler discretization of a continuous parabolic flow. For the pseudo-Calabi flow, the intrinsic flow is

$$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$

where $u_i$0 is the harmonic projection of $u_i$1 with respect to $u_i$2. Rubinstein’s discretization studied by Zhang is

$u_i$3

with fixed step size $u_i$4 (Zhang, 2023). Taking trace with respect to $u_i$5 yields the twisted cscK equation

$u_i$6

so constant scalar curvature metrics are fixed points.

At the potential level, if $u_i$7 and

$u_i$8

then each step solves the coupled elliptic system

$u_i$9

$u_i$0

with normalization such as $u_i$1 or $u_i$2 (Zhang, 2023). This realizes each iteration step as a complex Monge–Ampère/Laplace solve rather than an explicit curvature descent.

The principal structural results are existence, monotonicity, and convergence. There exists $u_i$3, depending only on $u_i$4 and $u_i$5, such that for any $u_i$6 the Ricci iteration has a unique solution at each step, and one can take $u_i$7 when the K-energy is bounded from below. Along the iteration,

$u_i$8

with equality at some step iff the sequence is stationary and $u_i$9 is cscK. If $(\varepsilon,\eta)$0 admits a cscK metric, then for any $(\varepsilon,\eta)$1 there exist biholomorphisms $(\varepsilon,\eta)$2 such that $(\varepsilon,\eta)$3 smoothly, where $(\varepsilon,\eta)$4 is cscK; in the unique cscK case, convergence is smooth without composing by automorphisms (Zhang, 2023).

In the Fano case, Darvas and Rubinstein analyze the Kähler–Ricci iteration

$(\varepsilon,\eta)$5

which for $(\varepsilon,\eta)$6 becomes the inverse Ricci step

$(\varepsilon,\eta)$7

In potentials, the Fano $(\varepsilon,\eta)$8-iteration is

$(\varepsilon,\eta)$9

and the paper proves smooth convergence modulo holomorphic automorphisms to a Kähler–Einstein metric whenever one exists (Darvas et al., 2017). The cscK algorithm in arbitrary Kähler classes and the KE iteration in $\{\ell_i\}$0 are thus closely related, but they target different fixed-point equations.

3. Triangulated surfaces and discrete conformal flows

For triangulated surfaces, a major line of work formulates discrete Ricci flow as an evolution of vertex-based conformal factors. In the unified framework of surface Ricci flow, a discrete surface is a triangular mesh $\{\ell_i\}$1 in background geometry $\{\ell_i\}$2. Each vertex carries a circle radius $\{\ell_i\}$3 and a scheme coefficient $\{\ell_i\}$4, each edge carries a conformal-structure coefficient $\{\ell_i\}$5, and the optimization variables are

$\{\ell_i\}$6

This single parameterization covers Thurston’s circle packing, tangential circle packing, inversive distance circle packing, discrete Yamabe flow, virtual radius circle packing, and mixed-type schemes (zhang et al., 2014).

The discrete Gauss curvature at a vertex is

$\{\ell_i\}$7

with $\{\ell_i\}$8 replacing $\{\ell_i\}$9 for boundary vertices. The mesh Ricci energy is

$2$0

and its gradient satisfies

$2$1

Accordingly, the Ricci flow is the negative gradient flow

$2$2

while Newton iteration uses

$2$3

with Hessian assembled either from face Hessians or, in Euclidean power-Delaunay settings, from dual-edge weights (zhang et al., 2014).

Convexity properties depend on the background geometry. In the Euclidean setting, the Ricci energy is convex on the subspace $2$4 for power-Delaunay meshes, and the Hessian is SPD there. In the hyperbolic setting, the Hessian is SPD on all $2$5. In the spherical setting, the energy is generally not convex, and the practical approach stated in the paper is to compute in Euclidean background and stereographically project to the sphere (zhang et al., 2014). This nonconvexity is one of the recurring caveats in discrete surface Ricci flow.

A hyperbolic variant replaces classical angle-deficit curvature by an area-normalized curvature. For a circle-packing metric $2$6 with edge lengths determined by

$2$7

the normalized discrete Gaussian curvature is

$2$8

Using the coordinate

$2$9

the normalized discrete Ricci flow is

$w:E\to(0,\infty)$0

The paper proves that a zero-curvature circle-packing metric exists if and only if the normalized discrete Ricci flow converges, and that the flow converges if the initial curvatures are all negative (Ge et al., 2015).

A different surface algorithm appears for discrete surfaces of revolution. There the discrete metric on each rotationally symmetric face is

$w:E\to(0,\infty)$1

and the normalized discrete Ricci flow is

$w:E\to(0,\infty)$2

with

$w:E\to(0,\infty)$3

The total area $w:E\to(0,\infty)$4 is constant in $w:E\to(0,\infty)$5, and the numerical flows approach explicitly parametrized discrete constant Gaussian curvature surfaces of revolution (Suda, 2023).

4. Simplicial, piecewise-flat, and surgery-based formulations

In higher-dimensional piecewise-flat geometry, discrete Ricci flow is built from Regge calculus. A simplicial geometry $w:E\to(0,\infty)$6 is determined by its edge lengths, curvature is concentrated on hinges $w:E\to(0,\infty)$7, and the circumcentric dual lattice $w:E\to(0,\infty)$8 provides orthogonal dual cells and hybrid volumes. The hinge deficit angle is

$w:E\to(0,\infty)$9

and the sectional curvature associated to a hinge is

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$0

where $$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$1 is the dual polygon area (Miller et al., 2013).

The Regge–Ricci flow equation is most naturally written on dual edges:

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$2

Tracing this to a primal edge $$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$3 yields

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$4

with averaging weighted by reduced hybrid volumes. The resulting system is sparse and local, though it requires repeated reconstruction of circumcenters, dual cells, hybrid volumes, and derivatives $$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$5 (Miller et al., 2013).

A related three-dimensional construction defines discrete scalar, sectional, and Ricci curvatures directly on piecewise-flat triangulations. For Voronoi or barycentric duals,

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$6

and the recommended edge-orthogonal sectional curvature is

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$7

The discrete Ricci curvature along an edge is then

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$8

and the piecewise-flat Ricci flow is

$$\frac{d}{dt}\ell_{uv}(t)=-\kappa(u,v;t)\,\ell_{uv}(t),$$9

or, in normalized form,

$$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$0

The paper reports convergence to smooth Ricci flow for $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$1, the $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$2-cylinder, Gowdy, and Nil-3 test geometries (Conboye et al., 2016).

A further specialization to piecewise-linear $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$3-geometries diagonalizes the flow through an explicitly constructed Forman-Ricci tensor. The governing edge ODE is

$$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$4

with $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$5 the average of endpoint scalar curvatures and $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$6 a discrete sectional curvature assembled from neighboring hinge curvatures and $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$7 weights (Alsing et al., 2017). In the axially symmetric neckpinch geometry studied there, cubic-spline-based adaptive mesh redistribution and surgery through a Type-1 neck pinch yield the expected Thurston decomposition into two lobes, each collapsing toward a $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$8-sphere geometry. This use of surgery is geometrically closer to Hamilton–Perelman flow than graph thresholding procedures, even though both are sometimes described as “Ricci flow with surgery.”

5. Weighted-graph Ricci flow

On graphs, discrete Ricci flow algorithms are built from edgewise curvatures derived from transport between local probability measures. In the undirected weighted-graph formulation used for network alignment, if $$\ell_{uv}^{(t+1)}=\ell_{uv}^{(t)}(1-\eta\,\kappa(u,v;t)),$$9 has edge weights $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$0 and weighted shortest-path metric $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$1, Ollivier–Ricci curvature on an edge $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$2–$$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$3 is

$$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$4

where $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$5 distributes mass $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$6 at $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$7 and $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$8 uniformly on neighbors, and $$\partial_t\omega_t=-\operatorname{Ric}(\omega_t)+HRic(\omega_t),$$9 is the $u_i$00-Wasserstein distance under the current weighted metric (Ni et al., 2018). The graph Ricci flow update is

$u_i$01

followed by renormalization

$u_i$02

After convergence, the weighted shortest-path distance under the final weights is the Ricci flow metric $u_i$03 (Ni et al., 2018).

For community detection, Ni et al. use Ollivier–Ricci curvature with lazy random walk and exponential neighbor weighting,

$u_i$04

and the discrete flow

$u_i$05

with renormalization to preserve total edge length. Negatively curved inter-community edges are stretched, positively curved intra-community edges shrink, and “surgery” removes large-weight edges to extract connected components as communities (Ni et al., 2019).

The continuous counterpart of normalized Ollivier flow on weighted graphs is

$u_i$06

with $u_i$07 preserved for all $u_i$08. For connected weighted graphs, positive initial weights summing to $u_i$09, and an injective Lipschitz $u_i$10, the paper proves existence and uniqueness for all $u_i$11 (Bai et al., 2020). On a finite star graph with at least three leaves and $u_i$12, the normalized flow converges to a constant-weighted star; on paths, leaf edges decrease and internal edges increase, and with edge contraction as surgery any weighted path converges to a path of length $u_i$13 (Bai et al., 2020).

A different graph algorithm freezes curvature between surgery events. In the piecewise-linear Ricci curvature flow on weighted graphs, the continuous flow is defined on each interval $u_i$14 by

$u_i$15

so

$u_i$16

At checkpoint times, an $u_i$17-surgery removes edges satisfying

$u_i$18

The framework supports arbitrarily selected homogeneous edge curvatures, including Ollivier, Lin–Lu–Yau, Forman, Menger, and Haantjes; it has global existence and uniqueness; and after the last surgery, each connected component has constant Ricci curvature across its edges (Ma et al., 21 May 2025). A key algorithmic distinction is that curvature is recomputed only at initialization and after surgeries, not at every iteration.

For directed weighted graphs, the balancing-factor formulation blends outflow and inflow at each node:

$u_i$19

with nodewise $u_i$20. Using the directed LLY curvature

$u_i$21

the normalized discrete update is

$u_i$22

with $u_i$23 preserved. The paper establishes existence and uniqueness for the continuous flow and gives an explicit Euler scheme for numerical computation (Bai et al., 24 Sep 2025).

Two further graph variants change the role of the flow rather than its basic transport geometry. In controlled network flow, the closed-loop equation is

$u_i$24

with Lyapunov-stable feedback

$u_i$25

used to regulate weights toward a target configuration and thereby alter entropy and robustness (Sandhu et al., 2019). In Ricci-Filtration for retrieval-augmented generation, the graph nodes are a query and retrieved chunks, all present edges are initialized with $u_i$26, normalized discrete Ricci flow is run for $u_i$27 iterations, and a chunk $u_i$28 is kept iff

$u_i$29

with default $u_i$30 on the mean-normalized scale (Qin et al., 13 Jun 2026).

6. Fixed points, convergence mechanisms, and applications

Across these constructions, fixed points are curvature-balanced states, but the meaning of “balanced” varies sharply. In Kähler geometry, fixed points are cscK or Kähler–Einstein metrics, and convergence is expressed in smooth topology, often modulo automorphisms (Zhang, 2023, Darvas et al., 2017). In triangulated-surface algorithms, the target is prescribed discrete Gauss curvature, typically constant, with convex Ricci energies and SPD Hessians providing rapid Newton convergence in Euclidean or hyperbolic regimes (zhang et al., 2014, Ge et al., 2015). In simplicial and piecewise-flat three-dimensional algorithms, the target is a piecewise-flat metric whose edgewise Ricci data approximate smooth Einstein or uniform-curvature behavior, with surgery used to pass singularities such as neckpinches (Alsing et al., 2017, Conboye et al., 2016). On graphs, equilibrium can mean uniformized curvature, constant curvature on each connected component after surgeries, or a task-dependent weight separation that reveals communities or stabilizes distances (Ni et al., 2018, Ma et al., 21 May 2025).

The main convergence mechanisms are likewise heterogeneous. Kähler iterations are governed by monotonicity of the Mabuchi K-energy or Ding functional and by elliptic regularity (Zhang, 2023, Darvas et al., 2017). Unified surface Ricci flow is governed by convexity of discrete Ricci energy and Hessian symmetry (zhang et al., 2014). Controlled graph flow uses Lyapunov functions, while normalized Ollivier flow on weighted graphs uses ODE well-posedness on the positive simplex (Sandhu et al., 2019, Bai et al., 2020). Piecewise-linear graph flow uses homogeneity plus finite surgery to force constant curvature within each component after the last surgery (Ma et al., 21 May 2025). This suggests that “discrete Ricci flow algorithm” should be understood less as a single numerical method than as a design pattern in which curvature determines a local metric update and a problem-specific functional, normalization, or surgery rule determines long-time behavior.

The application landscape is correspondingly broad. Weighted-graph Ricci flow has been used to define a robust Ricci flow metric for network alignment, with landmark-based coordinates and matching by Hungarian or greedy solvers (Ni et al., 2018). Geometric community detection uses thresholding of stretched edges after graph Ricci flow (Ni et al., 2019, Ma et al., 21 May 2025). In representation learning, one paper does not run an explicit curvature flow on graphs; rather, it evaluates whether neural feature graphs evolve like a Ricci flow through local Ricci evolution coefficients $u_i$31 and layer coefficients $u_i$32 (Hehl et al., 26 Sep 2025). In retrieval-augmented generation, Ricci-Filtration uses normalized discrete Ricci flow to remove noisy chunks before reranking (Qin et al., 13 Jun 2026). In complex differential geometry, Ricci iteration provides a discrete route to cscK and Kähler–Einstein metrics, including a new method of uniformization of the Riemann sphere (Zhang, 2023, Darvas et al., 2017).

Several recurring caveats are explicit in the literature. The long-time existence of the pseudo-Calabi flow itself is still an open question, whereas the corresponding Ricci iteration exists for all steps and decreases K-energy (Zhang, 2023). In the unified surface theory, spherical background energy is generally not convex (zhang et al., 2014). In graph settings, some algorithms recompute curvature at every iteration, while others obtain efficiency by event-driven caching and recomputation only after surgery (Ni et al., 2018, Ma et al., 21 May 2025). These are not superficial implementation choices; they encode different mathematical models of what it means to discretize Ricci flow.