  The existence of inter-dependence between multiple networks imparts an
additional scale of complexity to such systems often referred to as `network of
networks' (NON). We have investigated the robustness of NONs to random
breakdown of their components, as well as targeted attacks, as a function of
the relative proportion of intra- and inter-dependence among the constituent
networks. We focus on bi-layer networks with the two layers comprising
different number of nodes in general and where the ratio of intra-layer to
inter-layer connections, $r$, can be varied, keeping the total number of nodes
and overall connection density invariant. We observe that while the responses
of the different networks to random breakdown of nodes are similar, dominantly
intra-dependent networks ($r\ll1$) are robust with respect to attacks that
target nodes having highest degree but when nodes are removed on the basis of
highest betweenness centrality (CB), they exhibit a sharp decrease in the size
of the largest connected component (resembling a first order phase transition)
followed by a more gradual decrease as more nodes are removed (akin to a second
order transition). We also explore the role of layer size heterogeneity on
robustness, finding that for a given $r$ having layers comprising very
different number of nodes results in a bimodal degree distribution. For
dominantly inter-dependent networks, this results in the nodes of the smaller
layer becoming structurally central. Selective removal of these nodes, which
constitute a relatively small fraction of the network, leads to breakdown of
the entire system - making the inter-dependent networks even more fragile to
targeted attacks than scale-free networks having power-law degree distribution.
