I'm frankly dumbfounded. Unless I'm reading wrongly, @Mike has asked this question, and then provided his own answer.
To answer your question, though it hardly seems necessary since you knew your answer before asking your question, before "the church in the city of Rome" tried "to impose a tradition on other churches", whatever that means, other churches were querying the church of Rome for guidance. Pope St. Clement (+AD 96) wrote to the church of Corinth in response to complaints from its leadership. The presbyters and bishop of Corinth had been deposed and chased out of the city and they wrote to Rome asking its bishop to validate their authority as appointees of the apostles. And this happened perhaps while St. John was still even alive.
It is wise, when wondering how Rome started "imposing traditions other churches," to remember that while Rome had armies, the church of Rome had none. How, precisely, is the Bishop of Roman supposed to have imposed anything on anyone?
The appeal to Rome is always an appeal to moral authority. Even in the heyday of her power, the Church of Rome and the Papal States were by-and-large reliant on other nations' for their military protection. When patriarchs in Constantinople and bishops in North Africa ask for the support of the Pope, they are not asking for his armies - at his most militarized he had precious few - they are asking for his endorsement.