  Much of the complexity of social, biological, and engineered systems arises
from a network of complex interactions connecting many basic components.
Network analysis tools have been successful at uncovering latent structure
termed communities in such networks. However, some of the most interesting
structure can be difficult to uncover because it is obscured by the more
dominant structure. Our previous work proposes a general structure
amplification technique called HICODE that uncovers many layers of functional
hidden structure in complex networks. HICODE incrementally weakens dominant
structure through randomization allowing the hidden functionality to emerge,
and uncovers these hidden structure in real-world networks that previous
methods rarely uncover. In this work, we conduct a comprehensive and systematic
theoretical analysis on the hidden community structure. In what follows, we
define multi-layer stochastic block model, and provide theoretical support
using the model on why the existence of hidden structure will make the
detection of dominant structure harder compared with equivalent random noise.
We then provide theoretical proofs that the iterative reducing methods could
help promote the uncovering of hidden structure as well as boosting the
detection quality of dominant structure.
