  The concept of network efficiency, recently proposed to characterize the
properties of small-world networks, is here used to study the effects of errors
and attacks on scale-free networks. Two different kinds of scale-free networks,
i.e. networks with power law P(k), are considered: 1) scale-free networks with
no local clustering produced by the Barabasi-Albert model and 2) scale-free
networks with high clustering properties as in the model by Klemm and Eguiluz,
and their properties are compared to the properties of random graphs
(exponential graphs). By using as mathematical measures the global and the
local efficiency we investigate the effects of errors and attacks both on the
global and the local properties of the network. We show that the global
efficiency is a better measure than the characteristic path length to describe
the response of complex networks to external factors. We find that, at variance
with random graphs, scale-free networks display, both on a global and on a
local scale, a high degree of error tolerance and an extreme vulnerability to
attacks. In fact, the global and the local efficiency are unaffected by the
failure of some randomly chosen nodes, though they are extremely sensititive to
the removal of the few nodes which play a crucial role in maintaining the
network's connectivity.
