We study cascading failures in a system comprising interdependent networks/systems, in which nodes rely on other nodes both in the same system and in other systems to perform their function. The (inter-)dependence among nodes is modeled using a dependence graph, where the degree vector of a node determines the number of other nodes it can potentially cause to fail in each system through aforementioned dependency. In particular, we examine the impact of the variability and dependence properties of node degrees on the probability of cascading failures. We show that larger variability in node degrees hampers widespread failures in the system, starting with random failures. Similarly, positive correlations in node degrees make it harder to set off an epidemic of failures, thereby rendering the system more robust against random failures.