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Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications

Danillo Barros de Souza, Jonatas Teodomiro, Fernando A. N. Santos, Mengjun Ding, Weiqiang Sun, Mathieu Desroches, Jürgen Jost, Serafim Rodrigues

TL;DR

This work develops a locally computable Forman-Ricci curvature framework on Vietoris-Rips complexes by decomposing curvature contributions and proving a set-theoretical, incremental update mechanism. It establishes a Gauss-Bonnet–style relation linking local node-level curvatures to the global $F_d(C)$ and presents an efficient filtration-based algorithm that updates curvature when new faces are added. The authors validate the approach through data geometrization, converting tabular data into geometry-aware features across VR filtrations and applying UMAP for classification on synthetic point clouds and breast cancer datasets, with improved separation and robustness to noise. The method provides a geometry-centric complement to topological descriptors, offering insights for high-dimensional data analysis and potential extensions to other distance metrics beyond Euclidean.

Abstract

Discrete Forman-Ricci curvature (FRC) is an efficient tool that characterizes essential geometrical features and associated transitions of real-world networks, extending seamlessly to higher-dimensional computations in simplicial complexes. In this article, we provide two major advancements: First, we give a decomposition for FRC, enabling local computations of FRC. Second, we construct a set-theoretical proof enabling an efficient algorithm for the local computation of FRC in Vietoris-Rips (VR) complexes.Strikingly, this approach reveals critical information and geometric insights often overlooked by conventional classification techniques. Our findings open new avenues for geometric computations in VR complexes and highlight an essential yet under-explored aspect of data classification: the geometry underpinning statistical patterns.

Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications

TL;DR

This work develops a locally computable Forman-Ricci curvature framework on Vietoris-Rips complexes by decomposing curvature contributions and proving a set-theoretical, incremental update mechanism. It establishes a Gauss-Bonnet–style relation linking local node-level curvatures to the global and presents an efficient filtration-based algorithm that updates curvature when new faces are added. The authors validate the approach through data geometrization, converting tabular data into geometry-aware features across VR filtrations and applying UMAP for classification on synthetic point clouds and breast cancer datasets, with improved separation and robustness to noise. The method provides a geometry-centric complement to topological descriptors, offering insights for high-dimensional data analysis and potential extensions to other distance metrics beyond Euclidean.

Abstract

Discrete Forman-Ricci curvature (FRC) is an efficient tool that characterizes essential geometrical features and associated transitions of real-world networks, extending seamlessly to higher-dimensional computations in simplicial complexes. In this article, we provide two major advancements: First, we give a decomposition for FRC, enabling local computations of FRC. Second, we construct a set-theoretical proof enabling an efficient algorithm for the local computation of FRC in Vietoris-Rips (VR) complexes.Strikingly, this approach reveals critical information and geometric insights often overlooked by conventional classification techniques. Our findings open new avenues for geometric computations in VR complexes and highlight an essential yet under-explored aspect of data classification: the geometry underpinning statistical patterns.
Paper Structure (20 sections, 2 theorems, 21 equations, 22 figures, 2 tables, 3 algorithms)

This paper contains 20 sections, 2 theorems, 21 equations, 22 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Vietoris-Rips complexes from networks
  4. Discrete Forman-Ricci curvature
  5. Results
  6. Alternative formulation for local FRC computations
  7. A novel algorithm to efficiently compute FRC in a VR complex
  8. Application of Data Geometrization
  9. FRC computations on Synthetic point cloud data
  10. FRC computations on breast cancer datasets
  11. Conclusion
  12. Supplementary Material
  13. Theoretical Results
  14. Algorithms
  15. Main results and Demonstrations
  16. ...and 5 more sections

Key Result

Proposition 1

\newlabelprop:before_after0 Let $\alpha_i \in C_d$, $\alpha \in N^i$. Let $\mathrm{F}:C_d\rightarrow \mathbb{Z}$ and $\mathrm{F}^i$ be the FRC function as defined in eq:original_FRC. If $\alpha$ is parallel to $\alpha_i$, then $\mathrm{F}^i(\alpha)=\mathrm{F}^{i-1}(\alpha)-1$. Otherwise, we have $

Figures (22)

  • Figure 1: FRC computation on a VR complex as a function of the cutoff distance. The dashed-grey lines correspond to the cutoff values $0.7$, $1.0$, $1.4$ and $1.6$. The blue, green, yellow and red curves correspond to the average FRC for edges ($d=1$), triangles ($d=2$), tetrahedra ($d=3$) and pentahedra ($d=4$), respectively.
  • Figure 1: Example of an evolving simplicial complex (by adding a new face) and how it influences the FRC (the numerical values given in the figure) for edges (blue) and triangles (green). The new edge has $4$ neighbours, of which two are transverse (the neighbours sharing the same triangle) and $2$ parallel (The ones outside the triangle). The $1$-FRC of the new edge is the number of triangles containing the new edge plus the length of the boundary minus the number of parallel neighbours, i.e.,$1+2-2=1.$ When a new edge is added, the FRC increases by 2 units when a new triangle is created and decreases by 1 unit for the new neighbours outside the new triangle. Similarly, the new triangle has $3$ neighbours, where $1$ shares a tetrahedron and $2$ are parallel neighbours, therefore, the $2$-FRC of the new triangle is $1+3-2=2$. Similarly, for triangle faces, the FRC increases by 3 units when a new tetrahedron is created in the neighbourhood and decreases by 1 unit otherwise.
  • Figure 2: Computation of $d$-FRC for $d=1$ (first column), $d=2$ (middle column) and $d=3$ (third column), and fixed edge density $\rho=0.25$ in random geometric graphs with different box dimensions (see labels). The UMAP classification (bottom figures) used the Euclidean metric and the minimum distance of points $1.0$ (the highest allowed). We used this parameter to split away the points as much as possible and test the sensitivity to noise.
  • Figure 2: Computations of the average $d$-FRC for $d=1$ (left column), $d=2$ (middle column) and $d=3$ (right column) on Random geometric graphs with $n=100$ nodes and different box dimensions and densities (plot lines). The solid-coloured lines are the average, while the error bands are computed from the standard deviation. The box dimension classification can be better visualised in \ref{['fig:UMAP_FRC_RGG_all_plots']}.
  • Figure 3: FRC computation for different steps of Datasaurus dataset randomization. In the example, the algorithm \ref{['alg:randomizer']} was performed to disturb the original data (iteration 0) in a total of $9000$ iterations, which totalized $17469$ effective iterations. From these, we show the dataset changes for iterations $5000$, $10000$ and $15000$. Despite the randomization process that generates distinct geometry from VR complexes, the FRC presents robustness for low noise levels.
  • ...and 17 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Proof 1
  • Theorem A.1: Geometric Gauss-Bonnet Theorem
  • Proof 2