The eastern part of the Roman Empire came to be the most important part of the empire during the period of the Dominate largely because of two main factors.  One factor had to do with the problems being suffered by the city of Rome and the second had to do with the benefits of having a capital at Constantinople.

In the time before the Dominate, Rome and Italy were being badly pressured by barbarian forces.  In addition, civil war was rife.  The period from 235 to 284 AD was one of almost-constant civil war.  These problems greatly weakened the city of Rome.  It was left vulnerable to attacks by Germanic tribes.

At the same time, Constantinople had a great deal to recommend it as a center for the Roman Empire.  Constantinople was largely safe (at that time) from foreign invasion.  It was also located at a major crossroads between Europe and Asia.  This made it a very important center of trade.  Partly because of this, it was a fairly rich city.  For all of these reasons, the capital of the Roman Empire was moved to Constantinople and that region became the focal point of the Empire.