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Graph Multivector Persistence: A Unified Framework for Dynamic Systems

Donald Woukeng

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.

Graph Multivector Persistence: A Unified Framework for Dynamic Systems

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.
Paper Structure (24 sections, 14 theorems, 66 equations, 1 figure, 1 algorithm)

This paper contains 24 sections, 14 theorems, 66 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Graph Multivector Fields
  3. Generality on multivector fields
  4. Basic notions
  5. Purpose and Dynamical Interpretation
  6. From Combinatorial to Graph Multivector Fields
  7. Graph Multivector Fields
  8. Basic propositions
  9. Construction of Graph Multivector Fields
  10. Relation matrices
  11. Merging rules
  12. Algorithmic construction of the graph multivector field
  13. (1) Finiteness and termination.
  14. (2) Partition property.
  15. (3) Connectedness and local closedness.
  16. ...and 9 more sections

Key Result

Theorem 1

Alexandroff_ftop For a finite poset $(P,\leq)$ the family $\text{$\mathcal{T}$}_{\leq}$ of upper sets of $\leq$ is a $T_0$ topology on $P$. For a finite $T_0$ topological space $(X,\text{$\mathcal{T}$})$ the relation $x \leq_{\text{$\mathcal{T}$}} y$ defined by $x \in \operatorname{cl}_{\text{$\math

Figures (1)

  • Figure 1: Conley-Morse-graph for $\lambda=0.6$. Edges are drawn whenever $\operatorname{mo}(A_i)\cap A_j\neq\varnothing$. Each node is annotated with the pair $(\dim CH_0,\dim CH_1)$

Theorems & Definitions (39)

  • Definition 1: Finite poset
  • Theorem 1
  • Definition 2: Locally closed set
  • Definition 3: Combinatorial multivector field
  • Definition 4: Morse decomposition
  • Definition 5: Conley index lipinski2019conley
  • Definition 6: Graph poset
  • Definition 7: Local closedness in graphs
  • Definition 8: Graph multivector field
  • Proposition 1: Existence -- trivial multivector fields
  • ...and 29 more