We investigate certain structural properties of random interdependent networks. We start by studying a property known as $r$-robustness, which is a strong indicator of the ability of a network to tolerate structural perturbations and dynamical attacks. We show that random $k$-partite graphs exhibit a threshold for $r$-robustness, and that this threshold is the same as the one for the graph to have minimum degree $r$. We then extend this characterization to random interdependent networks with arbitrary intra-layer topologies. Finally, we characterize the algebraic connectivity of such networks, and provide an asymptotically tight rate of growth of this quantity for a certain range of inter-layer edge formation probabilities. Our results arise from a characterization of the isoperimetric constant of random interdependent networks, and yield new insights into the structure and robustness properties of such networks.