What are the challenges of empire, what is the usual result? or put into math terms:?

Diverse + Multiethnic = Hard to Unify

Empires generally covered large expansive territories across the world, these incorporated many diverse countries, and cultures under oppressive rules. Empires were often formed to demonstrate power and force of a nation often at the expense of citizens there. Due to the diversity and multi-ethnic nature of these territories, they were hard to unify and generally broke up due to the fact of being formed under duress.