How common is phase transition in NP-complete problems?
expert researchers in this area basically assert that phase transitions are a universal feature of NP complete problems although this has yet to be formulated/ proven rigorously and it is not yet widely regarded/ disseminated in the larger field (it emanates more from an empirical-oriented branch of study). its nearly an open conjecture. there is strong evidence. there are no plausible candidates for non phase-transition NP complete problems. here are two refs that support this pov:


Phase transitions in NP-complete problems:
a challenge for probability, combinatorics, and
computer science / Moore
Phase transition behavior / Walsh (ppt)


here is a rough sketch of the truth of the assertion. it has to do with P contained in NP complete. an NP complete problem/ language must have instances that are solvable in P time and others that are solvable in exponential (or at least superpolynomial-) time if P≠NP. but there must always be some way to "group" the P instances from the "non-P" instances. therefore there must also always be some "transition criteria" between the P and non-P instances. in short, maybe this phenomenon is intrisically coupled with P≠NP!

another rough argument: all NP complete problems are interchangeable via reductions. if a phase transition is found in a single one, it must be found in all of them.

more circumstantial evidence for this, more recently (~2010) it was shown the phase transition shows up for lower bounds on monotone circuits for clique-detection on random graphs.


Average case complexity of detecting cliques / Rossman 


full disclosure: Moshe Vardi has studied phase transitions particularly in SAT and has a contrasting more skeptical view in this talk/ video.


Phase transitions and computational complexity