  We analytically explore the scaling properties of a general class of nested
subgraphs in complex networks, which includes the $K$-core and the
$K$-scaffold, among others. We name such class of subgraphs $K$-nested
subgraphs due to the fact that they generate families of subgraphs such that
$...S_{K+1}({\cal G})\subseteq S_K({\cal G})\subseteq S_{K-1}({\cal G})...$.
Using the so-called {\em configuration model} it is shown that any family of
nested subgraphs over a network with diverging second moment and finite first
moment has infinite elements (i.e. lacking a percolation threshold). Moreover,
for a scale-free network with the above properties, we show that any nested
family of subgraphs is self-similar by looking at the degree distribution. Both
numerical simulations and real data are analyzed and display good agreement
with our theoretical predictions.
