Global properties of hereditary classes?
Hereditary properties are very "robust" in the following sense.

Noga Alon and Asaf Shapira showed that for any hereditary property ${\cal P}$, if a graph $G$ needs more than $\epsilon n^2$ edges to be added or removed in order to satisfy ${\cal P}$, then there is a subgraph in $G$, of size at most $f_{\cal P}(\epsilon)$, which does not satisfy ${\cal P}$.  Here, the function $f$ only depends on the property ${\cal P}$ (and not on the size of the graph $G$, for instance).  Erdős had made such a conjecture only about the property of $k$-colorability.

Indeed, Alon and Shapira prove the following stronger fact: given ${\cal P}$, for any $\epsilon$ in $(0,1)$, there are $N(\epsilon)$, $h(\epsilon)$ and $\delta(\epsilon)$ such that if a graph $G$ has at least $N$ vertices and needs at least $\epsilon n^2$ edges added/removed in order to satisfy ${\cal P}$, then for at least $\delta$ fraction of induced subgraphs on $h$ vertices, the induced subgraph violates ${\cal P}$.  Thus, if $\epsilon$ and the property ${\cal P}$ are fixed, in order to test if an input graph satisfies ${\cal P}$ or is $\epsilon$-far from satisfying ${\cal P}$, then one only needs to query the edges of a random induced subgraph of constant size from the graph and check if it satisfies the property or not.  Such a tester would always accept graphs satisfying ${\cal P}$ and would reject graphs $\epsilon$-far from satisfying it with constant probability.  Furthermore, any property that is one-sided testable in this sense is a hereditary property!  See the paper by Alon and Shapira for details.