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Detecting and characterizing community structure plays a crucial role in thestudy of networked systems. However, there is still a lack of understanding ofhow community structure affects the systems' resilience and stability. Here, wedevelop a framework to study the resilience of networks with communitystructure based on percolation theory. We find both analytically andnumerically that the interlinks (connections between the communities) affectthe percolation phase transition in a manner similar to an external field in aferromagnetic-paramagnetic spin system. We also study the universality class bydefining the analogous critical exponents $\delta$ and $\gamma$, and find thattheir values for various models and in real-world co-authors networks followfundamental scaling relations as in physical phase transitions. The methodologyand results presented here not only facilitate the study of resilience ofnetworks but also brings a fresh perspective to the understanding of phasetransitions under external fields.

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