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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.
Modern world builds on the resilience of interdependent infrastructures characterized as complex networks. Recently, a framework for analysis of interdependent networks has been developed to explain the mechanism of resilience in interdependent netwo
An indicator for presence of community structure in networks is suggested. It allows one to check whether such structures can exist, in principle, in any particular network, without a need to apply computationally cost algorithms. In this way we excl
Knowledge of time-variant functionality of real-world physical, social, and engineered networks is critical to the understanding of the resilience of networks facing external perturbations. The majority of existing studies, however, focus only on the
As two main focuses of the study of complex networks, the community structure and the dynamics on networks have both attracted much attention in various scientific fields. However, it is still an open question how the community structure is associate
Recent empirical studies suggest that heavy-tailed distributions of human activities are universal in real social dynamics [Muchnik, emph{et al.}, Sci. Rep. textbf{3}, 1783 (2013)]. On the other hand, community structure is ubiquitous in biological a