It makes sense for Peter's successors to follow him and considering that he chose to leave Antioch and go to Rome.
Indeed Peter was said by Jesus to have the keys to kingdom of heaven and that is why the Church authority follows Peter - in this case to Rome.
So if the people of Antioch had murdered Peter in Antioch then Antioch would be the highest authority of the Church?
Because Antiochians weren't murderers, they aren't the highest authority? How does that make sense?