The answer to your question can be found in Chapter 9 of The Outsiders. I have this book in electronic form, so I cannot give you a page number, but it is just a few pages before the end of Chapter 9. The answer is that Ponyboy’s brother Darry (his real name is Darrel) and a blond Soc named Paul Holden are the ones who start the rumble. Their past relationship is that they played football together in high school.
When the two sides have met and agreed to the rules for the rumble, no one is sure how to start things. At that point, Darry says he will fight anyone from the Soc’s side. At that point, the narrator tells us,
there was a slight stir in the faceless mob of Socs, and a husky blond guy stepped forward. He looked at Darry and said quietly, "Hello, Darrel."
The husky blond guy is Paul Holden. He knows Darry by name because he was the best running back on Darry’s high school football team (Darry was the captain of the team) and the two had been friends. Now, they will be the ones to start the fight. This is one way in which the author shows us the senselessness of the enmity between the Socs and the Greasers. These are young men who could and should be friends, but they are about to fight for no real reason other than their group identity.