Construction and Analysis of Random Networks with Explosive Percolation

The existence of explosive phase transitions in random (Erd\H os R'enyi-type) networks has been recently documented by Achlioptas et al.
[Science {\bf 323}, 1453 (2009)] via simulations. In this Letter we describe the underlying mechanism behind these first-order phase transitions and develop tools that allow us to identify (and predict) when a random network will exhibit an explosive transition. Several interesting new models displaying explosive transitions are also presented.