  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.
