  Hausdorff dimensions of level sets of generic continuous functions defined on
fractals can give information about the "thickness/narrow cross-sections'' of a
"network" corresponding to a fractal set, $F$. This lead to the definition of
the topological Hausdorff dimension of fractals. In this paper we continue our
study of the level sets of generic $1$-H\"older-$\alpha$ functions. While in a
previous paper we gave the initial definitions and established some properties
of these generic level sets, in this paper we provide numerical estimates in
the case of the Sierpi\'nski triangle. These calculations give better insight
and illustrate why can one think of these generic $1$-H\"older-$\alpha$ level
sets as something measuring "thickness/narrow cross-sections/conductivity'' of
a fractal "network". We also give an example for the phenomenon which we call
phase transition for $D_{*}(\alpha, F)$. This roughly means that for a certain
lower range { of $\alpha$s } only the geometry of $F$ determines $D_{*}(\alpha,
F)$ while for larger values the H\"older exponent, $\alpha$ also matters.
