  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.

Statistical Mechanics;Disordered Systems and Neural Networks