# Stable components and layers

## TL;DR

This paper introduces the concept of stable components and layers within component graphs derived from arrays of sets, particularly applied to Vietoris-Rips and Lesnick complexes, with scoring functions for analysis.

## Contribution

It defines stable components and layers in component graphs, providing a new framework for analyzing complex data structures in topological data analysis.

## Key findings

- Stable components are identified as path components of component graphs.
- Layers are decompositions of stable components within Lesnick complexes.
- Scoring functions are introduced for analyzing layers and stable components.

## Abstract

Component graphs $\Gamma_{0}(F)$ are defined for arrays of sets $F$, and in particular for arrays of path components for Vietoris-Rips complexes and Lesnick complexes. The path components of $\Gamma_{0}(F)$ are the {\it stable components} of the array $F$. The stable components for the system of Lesnick complexes $\{ L_{s,k}(X) \}$ for a finite data set $X$ decompose into layers, which are themselves path components of a graph. Combinatorial scoring functions are defined for layers and stable components.