Quantization of Ricci Curvature in Information Geometry

Carlos C. Rodriguez

cs.ITMar 9, 2026

Quantization of Ricci Curvature in Information Geometry

Authors: Carlos C. Rodriguez

Organizations: Department of Mathematics and Statistics, University at Albany, SUNY

Abstract

In 2004, while studying the information geometry of binary Bayesian networks (bitnets), the author conjectured that the volume-averaged Ricci scalar <R> computed with respect to the Fisher information metric is universally quantized to positive half-integers: <R> in (1/2)Z. This paper resolves the conjecture after 20 years. We prove it for tree-structured and complete-graph bitnets via a universal Beta function cancellation mechanism, and disprove it in general by exhibiting explicit loop counterexamples. We extend the program to Gaussian DAG networks, where a sign dichotomy holds: discrete bitnets have positive curvature, while Gaussian networks form solvable Lie groups with negative curvature.

Figures & tables

$n$ parents	$\tilde{C}_{n+1}$	$d$	$a_{n}$	$b_{n}$	$\langle R\rangle$
1	$\tilde{C}_{2}$	3	$3/2$	0	$3/2$
2	$\tilde{C}_{3}$	6	10	$1/2$	2
3	$\tilde{C}_{4}$	11	54	3	6
4	$\tilde{C}_{5}$	20	272	14	16
5	$\tilde{C}_{6}$	37	1008	60	$-272$

Table 1: Collapsing star

\tilde{C}_{n+1}

: volumes, coefficients and averaged Ricci scalars. Values for

n=1,2,3

were obtained with vTool [ 1 ] ;

n=4,5

are new SymPy computations.

Topology	Description	$n$ nodes	$d$	$\langle R\rangle$	Source
$\tilde{K}_{1}$	single node	1	1	0	trivial
$\tilde{K}_{2}=\tilde{L}_{2}=\tilde{E}_{2}$	chain/star 2-node	2	3	$3/2$	exact ( $R$ const)
$\tilde{L}_{3}=\tilde{E}_{3}$	chain/star 3-node	3	5	$5/2$	MC
$\tilde{L}_{n}=\tilde{E}_{n}$	chain/star $n$ -node	$n$	$2n{-}1$	$(2n-1)/2$	Thm. 3.2
$\tilde{C}_{3}$	V-structure (2 parents)	3	6	2	exact
$\tilde{C}_{4}$	3-parent collider	4	11	6	SymPy

Table 2: Summary: Bitnet topologies, dimensions, and averaged Ricci scalars.

Topology	$d$	$R$	Constant?	Notes
Single $\mathcal{N}(0,v)$	1	$0$	yes	trivial
Chain $X_{1}\to X_{2}$ ( $\mathbb{R}\times H^{2}$ )	3	$-2$	yes	SymPy
V-structure, indep. roots	5	$-5$	yes	SymPy
Chain $X_{1}\to X_{2}\to X_{3}$	5	$-5$	yes	SymPy
3-parent indep. star $\to X_{4}$	7	$-9$	yes	SymPy
Chain $X_{1}\to X_{2}\to X_{3}\to X_{4}$	7	$-9+v_{3}/E[X_{3}^{2}]$	no	chain coupling

Table 3: Gaussian DAG networks: dimensions and Ricci scalars (

v_{i}>0

a_{ij}\in\mathbb{R}

). “Indep. parents” means no two siblings share a common ancestor within the same node block.

Information Geometry

Information geometry studies the geometric structure of probability distributions and uses it to understand inference, learning, and optimization.

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Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure, and its graph-based analogues have recently gained prominence as practical tools for reweighting, pruning, and reshaping network geometry. We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature canonically induced by Sobolev transport geometry, which admits efficient evaluation via a tree-metric Sobolev structure on neighborhood measures. We establish two consistency behaviors that anchor SRC to classical transport curvature: (i) on trees endowed with the length measure, SRC recovers Ollivier-Ricci curvature (ORC) in the canonical W1 setting, and (ii) SRC vanishes in the Dirac limit, matching the flat case of measure-theoretic Ricci curvature. We demonstrate SRC as a reusable curvature primitive in two representative pipelines. We define Sobolev-Ricci Flow by replacing ORC with SRC in a Ricci-flow-style reweighting rule, and we use SRC for curvature-guided edge pruning aimed at preserving manifold structure. Overall, SRC provides a transport-based foundation for scalable curvature-driven graph transformation and manifold-oriented pruning.

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