The paper by Buldyrev et al. explores the catastrophic cascade of failures in interdependent networks, which are essential for understanding the robustness of real-world systems that rely on multiple interconnected networks. The authors introduce a model and analytical framework to study these networks, revealing surprising results that have significant implications for network design. In interdependent networks, the failure of nodes in one network can lead to failures in another network, potentially causing a complete fragmentation of all networks. The critical average degree below which both networks collapse is found to be $\langle k_c \rangle = 2.445$ for two Erdős-Rényi (ER) networks, compared to $\langle k_c \rangle = 1$ for a single ER network. Interestingly, while a broader degree distribution in a single network increases robustness to random failure, it makes interdependent networks more vulnerable to such failures. The authors use percolation theory and generating functions to solve the model analytically. They find that the critical fraction of nodes that must be removed to cause complete fragmentation is determined by the intersection of a straight line and a curve, which depends on the degree distributions of the networks. For ER networks, the critical fraction is given by a transcendental equation, and numerical simulations confirm the theoretical predictions. The paper also discusses the robustness of scale-free networks, where the critical fraction of nodes that must be removed remains finite even as the network size increases. This is in contrast to regular percolation, where the critical fraction tends to zero. The authors conclude that in interdependent networks, hubs, which are crucial for robustness in single networks, become vulnerable during cascading failures, making broad degree distributions more detrimental. Finally, the paper addresses finite size effects, showing that the width of the interval over which a complete fragmentation can occur scales with the square root of the network size. The average number of stages in the cascade of failures diverges proportionally to $\ln N/\sqrt{p-p_c}$, and the distribution of the number of stages has an exponential tail.The paper by Buldyrev et al. explores the catastrophic cascade of failures in interdependent networks, which are essential for understanding the robustness of real-world systems that rely on multiple interconnected networks. The authors introduce a model and analytical framework to study these networks, revealing surprising results that have significant implications for network design. In interdependent networks, the failure of nodes in one network can lead to failures in another network, potentially causing a complete fragmentation of all networks. The critical average degree below which both networks collapse is found to be $\langle k_c \rangle = 2.445$ for two Erdős-Rényi (ER) networks, compared to $\langle k_c \rangle = 1$ for a single ER network. Interestingly, while a broader degree distribution in a single network increases robustness to random failure, it makes interdependent networks more vulnerable to such failures. The authors use percolation theory and generating functions to solve the model analytically. They find that the critical fraction of nodes that must be removed to cause complete fragmentation is determined by the intersection of a straight line and a curve, which depends on the degree distributions of the networks. For ER networks, the critical fraction is given by a transcendental equation, and numerical simulations confirm the theoretical predictions. The paper also discusses the robustness of scale-free networks, where the critical fraction of nodes that must be removed remains finite even as the network size increases. This is in contrast to regular percolation, where the critical fraction tends to zero. The authors conclude that in interdependent networks, hubs, which are crucial for robustness in single networks, become vulnerable during cascading failures, making broad degree distributions more detrimental. Finally, the paper addresses finite size effects, showing that the width of the interval over which a complete fragmentation can occur scales with the square root of the network size. The average number of stages in the cascade of failures diverges proportionally to $\ln N/\sqrt{p-p_c}$, and the distribution of the number of stages has an exponential tail.