There are several synonyms for the phrase "running ahead." One commonly used term is "leading the pack," which implies that someone is at the forefront of a group. Another option is "breaking away," which suggests that someone is separating themselves from the rest of the group. "Forging ahead" is another option, which can be used to describe someone who is confidently and aggressively moving forward. A more casual phrase is "outpacing the others," which is commonly used in sports or competitions. "Taking the lead" is a more general phrase that can be used in a variety of contexts, and implies that someone is in charge or in front of others.