  Complex networks often have a modular structure, where a number of tightly-
connected groups of nodes (modules) have relatively few interconnections.
Modularity had been shown to have an important effect on the evolution and
stability of biological networks, on the scalability and efficiency of
large-scale infrastructure, and the development of economic and social systems.
An analytical framework for understanding modularity and its effects on network
vulnerability is still missing. Through recent advances in the understanding of
multilayer networks, however, it is now possible to develop a theoretical
framework to systematically study this critical issue. Here we study,
analytically and numerically, the resilience of modular networks under attacks
on interconnected nodes, which exhibit high betweenness values and are often
more exposed to failure. Our model provides new understandings into the
feedback between structure and function in real world systems, and consequently
has important implications as diverse as developing efficient immunization
strategies, designing robust large-scale infrastructure, and understanding
brain function.
