  In the present paper, we study the robustness of two-dimensional random
lattices (Delaunay triangulations) under attacks based on betweenness
centrality. Together with the standard definition of this centrality measure,
we employ a range-limited approximation known as $\ell$-betweenness, where
paths having more than $\ell$ steps are ignored. For finite $\ell$, the attacks
produce continuous percolation transitions that belong to the universality
class of random percolation. On the other hand, the attack under the full range
betweenness induces a discontinuous transition that, in the thermodynamic
limit, occurs after removing a sub-extensive amount of nodes. This behavior is
recovered for $\ell$-betweenness if the cutoff is allowed to scale with the
linear length of the network faster than $\ell\sim L^{0.91}$. Our results
suggest that betweenness centrality encodes information on network robustness
at all scales, and thus cannot be approximated using finite-ranged calculations
without losing attack efficiency.
