Phase transition in the detection of modules in sparse networks
  We present an asymptotically exact analysis of the problem of detecting communities in sparse random networks. Our results are also applicable to detection of functional modules, partitions, and colorings in noisy planted models. Using a cavity method analysis, we unveil a phase transition from a region where the original group assignment is undetectable to one where detection is possible. In some cases, the detectable region splits into an algorithmically hard region and an easy one. Our approach naturally translates into a practical algorithm for detecting modules in sparse networks, and learning the parameters of the underlying model. 
Aurelien Decelle, Florent Krzakala, Cristopher Moore and Lenka   Zdeborov\'a
cond-mat.stat-mech cs.LG cs.SI physics.soc-ph