  We propose a network metric, edge proximity, ${\cal P}_e$, which demonstrates
the importance of specific edges in a network, hitherto not captured by
existing network metrics. The effects of removing edges with high ${\cal P}_e$
might initially seem inconspicuous but are eventually shown to be very harmful
for networks. Compared to existing strategies, the removal of edges by ${\cal
P}_e$ leads to a remarkable increase in the diameter and average shortest path
length in undirected real and random networks till the first disconnection and
well beyond. ${\cal P}_e$ can be consistently used to rupture the network into
two nearly equal parts, thus presenting a very potent strategy to greatly harm
a network. Targeting by ${\cal P}_e$ causes notable efficiency loss in U.S. and
European power grid networks. ${\cal P}_e$ identifies proteins with essential
cellular functions in protein-protein interaction networks. It pinpoints
regulatory neural connections and important portions of the neural and brain
networks, respectively. Energy flow interactions identified by ${\cal P}_e$
form the backbone of long food web chains. Finally, we scrutinize the potential
of ${\cal P}_e$ in edge controllability dynamics of directed networks.
