  We study the detailed mechanism of the failure of scale-free networks under
intentional attacks. Although it is generally accepted that such networks are
very sensitive to targeted attacks, we show that for a particular type of
structure such networks surprisingly remain very robust even under removal of a
large fraction of their nodes, which in some cases can be up to 70%. The degree
distribution $P(k)$ of these structures is such that for small values of the
degree $k$ the distribution is constant with $k$, up to a critical value $k_c$,
and thereafter it decays with $k$ with the usual power law. We describe in
detail a model for such a scale-free network with this modified degree
distribution, and we show both analytically and via simulations, that this
model can adequately describe all the features and breakdown characteristics of
these attacks. We have found several experimental networks with such features,
such as for example the IMDB actors collaboration network or the citations
network, whose resilience to attacks can be accurately described by our model.
