Stable components and layers

Component graphs ${\mathrm{\Gamma }}_0(F)$ 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 ${\mathrm{\Gamma }}_0(F)$ 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.