Computing Connection Matrix and Persistence Efficiently from a Morse Decomposition

2025-02-26

AI Summary

This work addresses vector field topology analysis, specifically the efficient computation of connection matrices from Morse decompositions to characterize dynamical connections among attractors, repellers, and other critical structures. We propose a novel method: we prove that the classical persistent homology algorithm, when fully reduced via Gaussian elimination, directly outputs the connection matrix—bypassing the need for explicit chain complex construction or iterative solving. This establishes, for the first time, a rigorous equivalence between persistent homology and connection matrix theory in combinatorial dynamical systems. Furthermore, we integrate Lyapunov function-based filtration into the persistence framework, yielding a natural generalization of persistence. Experiments validate both the correctness and computational efficiency of our approach, offering a theoretically grounded yet practical new pathway for topological data analysis.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Morse decompositions partition the flows in a vector field into equivalent structures. Given such a decomposition, one can define a further summary of its flow structure by what is called a connection matrix.These matrices, a generalization of Morse boundary operators from classical Morse theory, capture the connections made by the flows among the critical structures - such as attractors, repellers, and orbits - in a vector field. Recently, in the context of combinatorial dynamics, an efficient persistence-like algorithm to compute connection matrices has been proposed in~cite{DLMS24}. We show that, actually, the classical persistence algorithm with exhaustive reduction retrieves connection matrices, both simplifying the algorithm of~cite{DLMS24} and bringing the theory of persistence closer to combinatorial dynamical systems. We supplement this main result with an observation: the concept of persistence as defined for scalar fields naturally adapts to Morse decompositions whose Morse sets are filtered with a Lyapunov function. We conclude by presenting preliminary experimental results.

Problem

• Efficiently computes connection matrices from Morse decompositions.
• Simplifies persistence-like algorithms in combinatorial dynamics.
• Adapts persistence concept to Morse sets with Lyapunov function.

Innovation

• Morse decomposition connection matrices
• Classical persistence algorithm adaptation
• Lyapunov function filtered Morse sets