If you are unfamiliar with the term, an 'empire of trust' is an idea coin by the historian Thomas Madden, to describe ad hoc empires built by the Roman Republic and the United States. Essentially, these are empires built by accident, through a web of alliances built by a militarily hegemonic but non-expansionist state that were sought simply to protect their homeland and allies, rather than expande their territory. Madden's presentation of this idea strays a bit too far into apologism, but the basic idea is sound, and lets accept the premise for the sake of discussion, and not worry about the flaws in his theory (though that would be a good discussion for another time).
Could the late Roman Empire/Byzantine Empire have done the same thing? I'm not referring to vassalizing the post-Roman states, demanding tribute, but forming genuine alliances of de jure equals with them, that eventually would entangle those states back into the Roman state?
Problems with this range from the relative power of the post-Roman states to the Empire - though that didn't stop the Republic from doing just this to Egypt - to the contemporary Roman's opposition to the idea that the barbarians should be treated in such terms.
Advantages to this approach are that the Late Empire has many more tools at its disposal to draw allies into their orbit than the Republic had. Basically, the Republic had two tools: They were militarily dominant, and regarded as trustworthy allies. Meanwhile, the Late Empire (I'm counting the east in this formulation) usually had the military advantage (again, when you count the east), but also had a vast economic advantage (again, counting the east), as well as cultural and religious advantages (with or without the east).