Graph Multivector Persistence: A Unified Framework for Dynamic Systems

TL;DR

This paper introduces a new combinatorial persistence invariant for finite weighted graphs, capturing dynamic changes across thresholds without relying on traditional homology, and proves its stability.

Contribution

It presents a novel combinatorial framework for persistent invariants of graphs based on multivector dynamics and Morse decompositions, independent of homology.

Findings

The persistence diagram is stable under perturbations of the relation matrix.

The method provides a topologically enriched invariant via Conley indices.

The framework is purely combinatorial, avoiding simplicial homology.

Abstract

We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a Morse decomposition given by its strongly connected components. As the threshold varies, these multivector fields form a monotone refinement family. We define the Morse persistence diagram by recording the birth and death of Morse sets along this filtration. The construction is purely combinatorial and does not rely on simplicial homology or persistence modules. We prove that the resulting persistence diagram is stable with respect to perturbations of the relation matrix in the sup norm. Each Morse set furthermore carries a combinatorial Conley index, yielding a topologically enriched invariant for multiscale graph structure.