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Persistent commutative algebra on graphs and hypergraphs

Faisal Suwayyid, Guo-Wei Wei

TL;DR

The paper develops a functorial, Tor-based notion of persistence for edge ideals of graphs and hypergraphs, extending Hochster's formula to a persistent setting and introducing persistent minimal primes. It builds a comprehensive persistent Betti theory that tracks syzygies and generators across filtrations, and it provides a prime-barcode perspective via minimal vertex covers. The framework generalizes to hypergraphs and facet ideals, preserving key tools like Betti splitting and the Betti shift, and demonstrates practical relevance through alignment-free genome classification and isomer discrimination. By bridging algebraic persistence with topological persistence, the work offers interpretable, multiscale invariants for evolving combinatorial structures with potential for broad data-driven applications.

Abstract

We introduce a persistent commutative algebra for studying the algebraic and combinatorial evolution of edge ideals of graphs and hypergraphs under filtration. Building on the Persistent Stanley--Reisner Theory (PSRT), we develop the notion of persistent edge ideals and analyze their graded Betti numbers across the filtration of graphs or hypergraphs. To enable this analysis, we establish a persistent extension of Hochster's formula, providing a functorial correspondence between algebraic and topological persistence. We further examine the behavior of Betti splittings in the persistent setting, proving a general inequality that extends the classical splitting result to the filtration of monomial ideals. Motivated by graph-theoretic interpretations, we introduce persistent minimal vertex covers, which encode the temporal structure of combinatorial dependencies within evolving graphs or hypergraphs. Applications to alignment-free genomic classification and molecular isomer discrimination demonstrate the interpretability and representatbility of persistent edge ideals as algebraic invariants, bridging combinatorial commutative algebra and data science.

Persistent commutative algebra on graphs and hypergraphs

TL;DR

The paper develops a functorial, Tor-based notion of persistence for edge ideals of graphs and hypergraphs, extending Hochster's formula to a persistent setting and introducing persistent minimal primes. It builds a comprehensive persistent Betti theory that tracks syzygies and generators across filtrations, and it provides a prime-barcode perspective via minimal vertex covers. The framework generalizes to hypergraphs and facet ideals, preserving key tools like Betti splitting and the Betti shift, and demonstrates practical relevance through alignment-free genome classification and isomer discrimination. By bridging algebraic persistence with topological persistence, the work offers interpretable, multiscale invariants for evolving combinatorial structures with potential for broad data-driven applications.

Abstract

We introduce a persistent commutative algebra for studying the algebraic and combinatorial evolution of edge ideals of graphs and hypergraphs under filtration. Building on the Persistent Stanley--Reisner Theory (PSRT), we develop the notion of persistent edge ideals and analyze their graded Betti numbers across the filtration of graphs or hypergraphs. To enable this analysis, we establish a persistent extension of Hochster's formula, providing a functorial correspondence between algebraic and topological persistence. We further examine the behavior of Betti splittings in the persistent setting, proving a general inequality that extends the classical splitting result to the filtration of monomial ideals. Motivated by graph-theoretic interpretations, we introduce persistent minimal vertex covers, which encode the temporal structure of combinatorial dependencies within evolving graphs or hypergraphs. Applications to alignment-free genomic classification and molecular isomer discrimination demonstrate the interpretability and representatbility of persistent edge ideals as algebraic invariants, bridging combinatorial commutative algebra and data science.

Paper Structure

This paper contains 12 sections, 11 theorems, 112 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded and Multigraded Betti Numbers
  4. Edge Ideals and the Independence Complex
  5. Persistent graded Betti numbers of modules and monomial ideals
  6. Persistent minimal primes for edge ideals
  7. Hypergraphs, edge ideals, and facet ideals
  8. Applications
  9. Application: Genome Classification via Persistent Edge Ideals
  10. Molecular Representation
  11. Discussion
  12. Conclusion

Key Result

Proposition 2.2

If $I\subseteq S$ is generated by squarefree monomials, then $I$ is radical; equivalently $\sqrt{I}=I$.

Figures (2)

  • Figure 1: Visualization of (a) the cis- isomer PubChem_CID643833 and (b) the trans- isomer PubChem_CID638186.
  • Figure 2: Comparison between (a) the cis- isomer and (b) the trans- isomer graded persistent Betti curves of edge ideals of the dichloroethene molecule derived from the Vietoris--Rips edge ideal filtration.

Theorems & Definitions (25)

  • Definition 2.1: Edge ideal
  • Proposition 2.2: Squarefree monomial ideals are radical
  • Corollary 2.3: Prime intersection decomposition, HerzogHibi2011
  • Definition 2.4: Vertex cover and minimal vertex cover
  • Theorem 2.5
  • proof
  • Definition 2.6: Independence complex
  • Proposition 2.7: Edge ideals as Stanley--Reisner ideals, VanTuyl2013_BeginnerGuideEdgeCoverIdeals
  • Theorem 2.8: Hochster's formula (multigraded form), bruns1998cohen
  • proof
  • ...and 15 more