How was the christian church organized by the 4th century

They say that Early Church organization copied the leadership pattern of the Roman Empire. 1. A "Deacon" led each local congregration. 2. A "Bishop" oversaw several Deacons 3."Archbishops" developed to see large Bishoprics 4. Eventually the Bishop of Rome became the unofficial leader of the church and gained the title of "Pope" 5. The first Bishop of Rome was Simon Peter, the apostle. How did they come up with Pope? It was simply a honor title that was given to all bishops of the Catholic church, but became, in time, to desinate the chief one of them...the leader of the Catholics...the head bishop. Hopefully that helped...