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Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes

Danillo Barros de Souza, Jonatas T. S. da Cunha, Fernando A. N. Santos, Jürgen Jost, Serafim Rodrigues

TL;DR

High-order Forman-Ricci curvature is powerful for capturing higher-order structure in networks but computationally expensive. The authors propose a set-theoretic formulation based on node neighbourhoods to enable efficient computation of $F_d(\alpha)$ for simplicial complexes, particularly Vietoris-Rips type where all simplices fill after the $1$-skeleton is present. They derive explicit expressions for neighborhoods ($N_\alpha$, $T_\alpha$, $P_\alpha$) and curvature, introduce FastForman with three variants, and demonstrate substantial improvements in time and memory compared to Hodgelaplacians and GeneralisedFormanRicci in benchmarks. This framework paves the way for scalable geometry-aware analysis in high-dimensional data and for extensions to cell complexes.

Abstract

Forman-Ricci curvature (FRC) is a potent and powerful tool for analysing empirical networks, as the distribution of the curvature values can identify structural information that is not readily detected by other geometrical methods. Crucially, FRC captures higher-order structural information of clique complexes of a graph or Vietoris-Rips complexes, which is not readily accessible to alternative methods. However, existing FRC platforms are prohibitively computationally expensive. Therefore, herein we develop an efficient set-theoretic formulation for computing such high-order FRC in simplicial complexes. Significantly, our set theory representation reveals previous computational bottlenecks and also accelerates the computation of FRC. Finally, We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. We envisage that FastForman will be used in Topological and Geometrical Data analysis for high-dimensional complex data sets. Moreover, our development paves the way for future generalisations towards efficient computations of FRC on cell complexes.

Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes

TL;DR

High-order Forman-Ricci curvature is powerful for capturing higher-order structure in networks but computationally expensive. The authors propose a set-theoretic formulation based on node neighbourhoods to enable efficient computation of for simplicial complexes, particularly Vietoris-Rips type where all simplices fill after the -skeleton is present. They derive explicit expressions for neighborhoods (, , ) and curvature, introduce FastForman with three variants, and demonstrate substantial improvements in time and memory compared to Hodgelaplacians and GeneralisedFormanRicci in benchmarks. This framework paves the way for scalable geometry-aware analysis in high-dimensional data and for extensions to cell complexes.

Abstract

Forman-Ricci curvature (FRC) is a potent and powerful tool for analysing empirical networks, as the distribution of the curvature values can identify structural information that is not readily detected by other geometrical methods. Crucially, FRC captures higher-order structural information of clique complexes of a graph or Vietoris-Rips complexes, which is not readily accessible to alternative methods. However, existing FRC platforms are prohibitively computationally expensive. Therefore, herein we develop an efficient set-theoretic formulation for computing such high-order FRC in simplicial complexes. Significantly, our set theory representation reveals previous computational bottlenecks and also accelerates the computation of FRC. Finally, We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. We envisage that FastForman will be used in Topological and Geometrical Data analysis for high-dimensional complex data sets. Moreover, our development paves the way for future generalisations towards efficient computations of FRC on cell complexes.
Paper Structure (11 sections, 12 theorems, 34 equations, 6 figures, 4 algorithms)

This paper contains 11 sections, 12 theorems, 34 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Network and higher-order network fundamentals
  3. Abstract simplicial complex
  4. High-order Forman-Ricci curvature
  5. Results
  6. Set-theoretic representation of Forman-Ricci curvature
  7. Computation and Benchmark of Novel Algorithms
  8. Conclusion
  9. Appendix
  10. Examples
  11. Demonstrations

Key Result

Theorem 6.1

\newlabeltheo:C_d0 Let $C_d$ be the set of all $d$-faces of a simplicial complex, as defined in sec:network_background. Then,

Figures (6)

  • Figure 1: Example of a simplicial complex and its faces for $d\in \{0,1,2,3\}$. In this example, we have $12$ nodes ($0$-faces), $13$ edges ($1$-faces), $5$ triangles ($2$-faces) and $1$ tetrahedron ($3$-faces).
  • Figure 1: Example of how the algorithm detects face neighbours from node neighbourhood for $2$-faces (triangles). The $2$-face $\alpha$ is enhanced in orange, whilst its boundary is drawn in red. According to the algorithm, the number of $3$-faces containing $\alpha$ in its boundary ($|H_{\alpha}|$) coincides with the number of common neighbours of the nodes in $\alpha$ ($|\pi_\alpha|$). Also, the number of neighbours per boundary is counted by the size of intersections of node neighbourhoods in the boundary. In this example, $\mathrm{F}_2(\alpha)=4\cdot|\{6\}|+3-(|\{3,6\}|+|\{6\}|+|\{6\}|)=3.$ The same output is reached by using \ref{['eq:forman_d_cells']}: $\mathrm{F}_2(\alpha)=1+3-1=3.$
  • Figure 1: Neighborhood condition depicted, for $d\in\{1,2,3\}$. For all dimensions, the neighbouring faces $\alpha$ and $\alpha'$ are drawn in blue and yellow, respectively, whilst the common boundary ($\alpha\cap\alpha'$) is represented in green. The presence (or absence) of the red dashed edge in the simplicial complex defines whether $\alpha$ and $\alpha'$ are transverse or parallel. In all cases, the intersection $\gamma:=\alpha \cap \alpha '$ is a $(d-1)$-face such that $\gamma<\alpha, \alpha'$, which guarantees that for $\alpha$ and $\alpha'$ to be neighbours, it is sufficient to reach the faceneighbourhood condition $1.$ is reached. This result is a consequence of \ref{['theo:2implies1']}.
  • Figure 2: Average number of $d$-faces for the set of $50$ point cloud data, and $d_{\max}\in\{2,5\}.$
  • Figure 3: Benchmark for computing Forman-Ricci curvature by using HodgeLaplacians, GeneralisedFormanRicci and FastForman With methods A, B and C, in comparison with the faces computation by using the Gudhi algorithm, for $n=50$ and $d_{\max}\in\{2,5\}$. We display the result in a log scale to clarify the analysis.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Remark 3.1
  • Example 6.1: Undirected Simple Graph
  • Example 6.2: Neighborhood of a node
  • Example 6.3: $d$-faces and Simplicial Complex
  • Example 6.4: Node neighbourhood of a face
  • Example 6.5: Boundary of a cell
  • Example 6.6: face neighborhood
  • Example 6.7: Face neighborhood condition
  • Example 6.8: Forman-Ricci curvature
  • Theorem 6.1
  • ...and 23 more