Anyway, I'm reading the section about "The Growing Importance of Rome"1 right now, and Walker is acknowledging a sort of historical primacy of Rome.
Even antedating what he identifies as the point when the Catholic Church really began structurally crystallizing, there does seem to be a recognized authority carried by the congregation in Rome, where "Paul and Peter died":
Even before the close of the first century Clement, writing anonymously to the Corinthians in the name of the whole Roman congregation ([c.] 93-97), spoke as for those who expected to be obeyed. The tone, if brotherly, was big-brotherly.Later in the paragraph, he mentions Irenaeus of Lyons assuming that all churches needed to agree with the Roman church.
I found it interesting.
1. Walker, Williston. A History of the Christian Church. 3rd ed. New York: Scribner, 1970. All quotes come from pp. 60-61.