  We consider a multi-layer network with two layers, $\mathcal{L}_{1}$,
$\mathcal{L}_{2}$. Their intra-layer topology shows a scale-free degree
distribution and a core-periphery structure. A nested structure describes the
inter-layer topology, i.e., some nodes from $\mathcal{L}_{1}$, the generalists,
have many links to nodes in $\mathcal{L}_{2}$, specialists only have a few.
This structure is verified by analyzing two empirical networks from ecology and
economics. To probe the robustness of the multi-layer network, we remove nodes
from $\mathcal{L}_{1}$ with their inter- and intra-layer links and measure the
impact on the size of the largest connected component, $F_{2}$, in
$\mathcal{L}_{2}$, which we take as a robustness measure. We test different
attack scenarios by preferably removing peripheral or core nodes. We also vary
the intra-layer coupling between generalists and specialists, to study their
impact on the robustness of the multi-layer network. We find that some
combinations of attack scenario and intra-layer coupling lead to very low
robustness values, whereas others demonstrate high robustness of the
multi-layer network because of the intra-layer links. Our results shed new
light on the robustness of bipartite networks, which consider only inter-layer,
but no intra-layer links.
