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 $ellsim L^{0.91}$. 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.
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