The Calabi flow with prescribed curvature on finite graphs

Abstract: In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent characterization of the problem, namely, the solution to the Calabi flow exists globally in time and converges if and only if there exists a weight function that realizes the prescribed curvature. In particular, for constant curvature weights, we prove that the solution to the Calabi flow exists globally in time and converges under certain topological conditions.

• The paper establishes that a weight function realizing prescribed curvature exists if and only if the modified Calabi flow converges globally on finite graphs.
• It employs discrete analogues of classical curvature flows, introducing fractional and p-th Calabi flows to address prescribed curvature under high girth constraints.
• The paper proves that canonical edge weights are unique up to scaling and can be computed constructively, offering practical algorithms for network geometry applications.

The Calabi Flow with Prescribed Curvature on Finite Graphs: An Analytical Overview

Introduction and Motivation

The study examines a discrete analogue of the Calabi flow, originally formulated for smooth Riemannian surfaces, in the context of finite graphs. Specifically, it addresses flows defined with respect to the Lin-Lu-Yau curvature on finite connected graphs of girth at least six. The central aim is to resolve the prescribed curvature problem: given a target curvature assignment on the edges of a graph, under what conditions does there exist a weight function on the edges realizing this assignment, and can such a weight function be constructed dynamically via a curvature flow?

This direction is an extension of the discrete Ricci and Calabi flows for surfaces [Chow-Luo (2003)], and responds to prior work by Lin and Liu on Ricci-type flows for graphs [Lin-Liu, (Lin et al., 11 Mar 2026)], broadening both the theoretical and applied scope of combinatorial curvature flows in discrete geometry and network science.

Curvature and Flow Definitions on Graphs

Lin-Lu-Yau Curvature under Girth Constraints

The Lin-Lu-Yau curvature, a modification of the Ollivier-Ricci curvature, measures the deviation between the Wasserstein distance and the combinatorial distance on a weighted graph [Lin, Lu, Yau 2011]. On graphs with girth at least six (i.e., no cycles of length ≤ 5), the pairwise curvature $\kappa_e$ for each edge $e$ with endpoints $x$ and $y$ simplifies to:

$$\kappa_e = 2\omega_e \left( \frac{1}{m(x)} + \frac{1}{m(y)} \right) - 2$$

where $\omega_e$ is the edge weight and $m(x)$ is the sum of weights of edges incident to $x$.

The total curvature satisfies a discrete Gauss-Bonnet-type identity:

$\sum_{e\in E} \kappa_e = 2(|V| - |E|).$

Discrete Calabi Flow and Its Variants

Building on smooth analogues, the authors define a discrete Calabi flow on the edge weights, parameterized by $r_i = \ln \omega_i$. The basic (unmodified) Calabi flow reads as

$e$0

where the discrete Laplacian $e$1 is defined via the Jacobian of the curvature map.

A modified Calabi flow with prescribed curvature $e$2 is given by

$e$3

Further generalizations include:

• Fractional Calabi flows, employing fractional matrix powers of the Laplacian, parametrized by $e$4: $e$5.
• $e$6-th Calabi flows, using a discrete $e$7-Laplacian appropriate for nonlinear flows and defined as

$e$8

All flows operate on graphs of girth at least six, ensuring key simplifications and rigidity in curvature-weight correspondence.

Main Theoretical Results

Equivalence of Prescribed Curvature Realizability and Flow Convergence

The core theorem states that, for a finite connected graph with girth at least six and prescribed curvature assignment $e$9 subject to the Gauss-Bonnet constraint, the following are equivalent:

• There exists a weight function yielding the prescribed curvature.
• The modified Calabi flow exists globally and converges exponentially to the realizing weight.
• The fractional and $x$0-th Calabi flows (for any $x$1, $x$2) exist globally and converge (exponentially in the first two cases, plain convergence for $x$3).

This establishes both necessary and sufficient dynamical conditions for solving prescribed curvature problems in this discrete setting.

Rigidity and Uniqueness

The system exhibits strong rigidity: the weight vector realizing a given edge curvature is unique up to scaling. This matches the continuous surface theory, where curvature-determining flows often yield unique canonical metrics within a conformal (or, here, multiplicative) class.

Conditions for Constant Curvature Weights

The existence of a weight function with constant Lin-Lu-Yau curvature is characterized by a combinatorial inequality on edge-vertex ratios across subgraphs. This parallels obstruction phenomena in classical uniformization and highlights new combinatorial invariants influencing discrete curvature flows.

Analytical Tools and Properties

• The Jacobian matrix of the curvature map is symmetric, positive semi-definite, and of rank $x$4, with nullspace corresponding to global scaling.
• The discrete Calabi energy functions are naturally convex, ensuring Lyapunov stability for the associated flows.
• The flows are strictly confined to an affine hyperplane, reflecting the conservation of a global "mass" parameter due to the nullspace structure.

Numerical and Quantitative Claims

• Exponential convergence is established for the modified and fractional Calabi flows: given realizability of the prescribed curvature, the distance to the target decays as $x$5 for some uniform $x$6.
• Only plain (non-exponential) convergence is proved for the $x$7-th Calabi flows with $x$8.
• The equivalences and sufficient-combinatorial conditions are both necessary and sharp.

Implications and Future Directions

The results constitute a discrete uniformization theorem for a class of graphs with sufficiently high girth, unifying previous developments in combinatorial Ricci and Calabi flows for surfaces and graphs. Practically, these flows offer constructive algorithms for finding canonical edge weightings matching arbitrary prescribed curvature data, relevant in spectral graph theory, discrete geometry, and the analysis of network robustness.

From a theoretical perspective, the classification and dynamics of these flows raise new questions about geometric structures on graphs with faces (lower girth), extensions to infinite graphs, random graph ensembles, and effective geometric invariants in complex networks.

Future developments could explore:

• Relaxing the girth constraint, possibly via alternative definitions of discrete curvature less sensitive to short cycles.
• Algorithmic optimization and numerical integration of the discrete Calabi flows.
• Applications to discrete geometric data analysis, including embedding networks in spaces of curvature.
• Connections to optimal transport on graphs, leveraging the Wasserstein foundations of Lin-Lu-Yau curvature.

Conclusion

This paper develops and analyzes discrete analogues of the Calabi flow with prescribed curvature for edge-weighted graphs of girth at least six, using the Lin-Lu-Yau curvature as a foundation. The equivalence between curvature realizability and dynamical convergence is rigorously established for a spectrum of combinatorial Calabi flows, solidifying the links between curvature prescription, geometric flows, and discrete graph geometry. The results consolidate and extend prior work on Ricci flows in discrete geometry, yielding both practical algorithms and novel perspectives on the combinatorial foundations of curvature flows on networks.