  Here we report a comprehensive analysis of the robustness of five
high-quality real-world complex weighted networks to errors and attacks of
nodes and links. We analyze C. Elegans, Cargo ship, E. Coli, Us Airports and
Human brain real-world complex weighted networks. We use measures of the
network damage conceived for a binary (e.g. largest connected cluster LCC, and
binary efficiency Effbin) or a weighted network structure (e.g. the efficiency
Eff, and the total energy-information En). We find that removing a very small
fraction of nodes and links with respectively higher strength and weight
triggers an abrupt collapse of the weighted functioning measures while measures
that evaluate the binary-topological connectedness are almost unaffected. These
findings unveil a problematic response-state of the real-world complex networks
where the attack of a small fraction of nodes-links returns these systems in a
connected but highly inefficient state. Our findings unveil how the robustness
may be heavily overestimated focusing on the connectedness of the components
only. Last, to understand how the networks robustness is generally affected by
link weights heterogeneity, we randomly assign link weights over the
topological structure of the real-world networks. We find that highly
heterogeneous networks experienced a faster efficiency decrease under
nodes-links removal: i.e. the robustness of the real-world complex networks
against both random than attack is negatively correlated with links weight
heterogeneity.
