Persistent commutative algebra on graphs and hypergraphs

TL;DR

This paper develops a new algebraic framework called persistent commutative algebra to analyze the evolution of edge ideals in graphs and hypergraphs, linking algebraic invariants with topological persistence for applications in data science.

Contribution

It introduces persistent edge ideals, extends Hochster's formula to the persistent setting, and explores the behavior of Betti splittings and minimal vertex covers over graph filtrations.

Findings

Established a persistent extension of Hochster's formula.

Proved a general inequality for Betti splittings in filtrations.

Applied the theory to genomic classification and molecular isomer discrimination.

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.