Resilience of networks with community structure behaves as if under an external field

Detecting and characterizing community structure plays a crucial role in the study of networked systems. However, there is still a lack of understanding of how community structure affects the systems' resilience and stability. Here, we develop a framework to study the resilience of networks with community structure based on percolation theory. We find both analytically and numerically that the interlinks (connections between the communities) affect the percolation phase transition in a manner similar to an external field in a ferromagnetic-paramagnetic spin system. We also study the universality class by defining the analogous critical exponents $\delta$ and $\gamma$, and find that their values for various models and in real-world co-authors networks follow fundamental scaling relations as in physical phase transitions. The methodology and results presented here not only facilitate the study of resilience of networks but also brings a fresh perspective to the understanding of phase transitions under external fields.