  It appeared recently that the underlying degree distribution of networks may
play a crucial role concerning their robustness. Empiric and analytic results
have been obtained, based on asymptotic and mean-field approximations. Previous
work insisted on the fact that power-law degree distributions induce high
resilience to random failure but high sensitivity to attack strategies, while
Poisson degree distributions are quite sensitive in both cases. Then, much work
has been done to extend these results.
  We aim here at studying in depth these results, their origin, and
limitations. We review in detail previous contributions and give full proofs in
a unified framework, and identify the approximations on which these results
rely. We then present new results aimed at enlightening some important aspects.
We also provide extensive rigorous experiments which help evaluate the
relevance of the analytic results.
  We reach the conclusion that, even if the basic results of the field are
clearly true and important, they are in practice much less striking than
generally thought. The differences between random failures and attacks are not
so huge and can be explained with simple facts. Likewise, the differences in
the behaviors induced by power-law and Poisson distributions are not as
striking as often claimed.
