  Network robustness against attacks has been widely studied in fields as
diverse as the Internet, power grids and human societies. Typically, in these
studies, robustness is assessed only in terms of the connectivity of the nodes
unaffected by the attack. Here we put forward the idea that the connectivity of
the affected nodes can play a crucial role in properly evaluating the overall
network robustness and its future recovery from the attack. Specifically, we
propose a dual perspective approach wherein at any instant in the network
evolution under attack, two distinct networks are defined: (i) the Active
Network (AN) composed of the unaffected nodes and (ii) the Idle Network (IN)
composed of the affected nodes. The proposed robustness metric considers both
the efficiency of destroying the AN and the efficiency of building-up the IN.
We show, via analysis of both prototype networks and real world data, that
trade-offs between the efficiency of Active and Idle network dynamics give rise
to surprising crossovers and re-ranking of different attack strategies,
pointing to significant implications for decision making.
