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Persistent homology of Morse decomposition in Markov chains based on combinatorial multivector fields

Donald Woukeng

TL;DR

The paper develops a persistence framework for Morse decompositions in Markov chains via combinatorial multivector fields, enabling topological tracking of recurrence structures as transition probabilities vary. It constructs M-fields directly from transition matrices, derives Morse decompositions whose Morse sets merge monotonically with increasing threshold $ extgamma$, and defines persistence diagrams indexed by birth/death times and Conley-index-derived topological data. A stability theorem is proven: if $ orm{P-P'}_ ext{$ ext∞$}<oldsymbol{\delta}$, then $d_B(D(P),D(P'))<Coldsymbol{\delta}$, ensuring robust persistence under small perturbations. An explicit three-state example illustrates multivector-field construction, Morse-set evolution, and persistence diagrams, highlighting the method's capacity to classify recurrent dynamics in finite Markov systems without relying on metric geometry.

Abstract

In this paper, we introduce a novel persistence framework for Morse decompositions in Markov chains using combinatorial multivector fields. Our approach provides a structured method to analyze recurrence and stability in finite-state stochastic processes. In our setting filtrations are governed by transition probabilities rather than spatial distances. We construct multivector fields directly from Markov transition matrices, treating states and transitions as elements of a directed graph. By applying Morse decomposition to the induced multivector field, we obtain a hierarchical structure of invariant sets that evolve under changes in transition probabilities. This structure naturally defines a persistence diagram, where each Morse set is indexed by its topological and dynamical complexity via homology and Conley index dimensions.

Persistent homology of Morse decomposition in Markov chains based on combinatorial multivector fields

TL;DR

The paper develops a persistence framework for Morse decompositions in Markov chains via combinatorial multivector fields, enabling topological tracking of recurrence structures as transition probabilities vary. It constructs M-fields directly from transition matrices, derives Morse decompositions whose Morse sets merge monotonically with increasing threshold , and defines persistence diagrams indexed by birth/death times and Conley-index-derived topological data. A stability theorem is proven: if ext∞, then , ensuring robust persistence under small perturbations. An explicit three-state example illustrates multivector-field construction, Morse-set evolution, and persistence diagrams, highlighting the method's capacity to classify recurrent dynamics in finite Markov systems without relying on metric geometry.

Abstract

In this paper, we introduce a novel persistence framework for Morse decompositions in Markov chains using combinatorial multivector fields. Our approach provides a structured method to analyze recurrence and stability in finite-state stochastic processes. In our setting filtrations are governed by transition probabilities rather than spatial distances. We construct multivector fields directly from Markov transition matrices, treating states and transitions as elements of a directed graph. By applying Morse decomposition to the induced multivector field, we obtain a hierarchical structure of invariant sets that evolve under changes in transition probabilities. This structure naturally defines a persistence diagram, where each Morse set is indexed by its topological and dynamical complexity via homology and Conley index dimensions.

Paper Structure

This paper contains 32 sections, 7 theorems, 31 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Overview of the result
  4. Preliminaries
  5. Relations and Posets
  6. Topological Spaces
  7. Combinatorial multivector fields
  8. Markov Chains
  9. Definition of a Markov Chain
  10. Graph Representation of a Markov Chain
  11. Morse decomposition for Markov chains
  12. Multivector field construction
  13. Morse Decomposition in Multivector Fields
  14. Stability of Persistence diagrams for Morse decomposition
  15. Persistence Diagram for Morse Sets
  16. ...and 17 more sections

Key Result

Theorem 1

Let $P$ and $P'$ be two transition matrices of a Markov process with n states such that: Then, the bottleneck distance between their Morse set persistence diagrams satisfies: where $C$ depends only on the structure of the Markov process.

Figures (1)

  • Figure 1: Persistence diagram of the Morse sets for $P$

Theorems & Definitions (13)

  • Theorem 1: Stability of Morse Set Persistence Diagrams
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2
  • Example 1
  • Remark 1: Persistence of Morse Sets
  • Remark 2
  • Theorem 3
  • Remark 3
  • Definition 1
  • ...and 3 more