Yes they did, but the Roman Empire doesn't qualify as points of light. The core of the empire, until its later years of course, was nearly free of violence. Rome practiced a system of hegemony to protect itself. In a very simple sense, there were three circles. The innermost circle was the Roman empire, which was free of disorder; this was seen to by the Roman Legions, which were posted in locations most likely to suffer rebellion (Jerusalem had one, Iberia had one, I believe England had two). Surrounding the empire were client states (this is the hegemony). Those client states were capable of Romanization, but the emperors following Augustus generally followed a policy of leaving the client states intact. They served as a buffer between Rome and the non-Romanizable barbarians beyond them. Romanization, by the way, was largely determined by how urban a region was. The client state didn't attack Rome because they recognized the threat of Rome's army and saw the benefit of peaceful trade. The barbarians saw neither. Rome kept those client states healthy and stable through a system of fairly substantial gifts to their rulers. The final circle was Rome's enemies. These were barbarians to the north and south and the Parthian empire to the east.
Thus Rome and, to a lesser degree, its client states were quite not a PoL system. The empire was a heavily civilized and heavily regulated society, and largely a peaceful one.
Now, the Roman empire by the time it began to crumble was definitely a different situation. However, the argument breaks down there as well because those regions which had degenerated to PoL status no longer saw any Roman officials, and this includes tax collectors, while those that still saw tax collectors, though not perfectly safe, were still not at constant threat from barbarians or other hostile armies.