1. an advantage, an element of aggression or directness
have a bit of an edge over your rivals
If you 'have an edge' on or over someone, you have an advantage or you are more likely to succeed while they fail. A company that successfully corners the market with their leading product could be described as having 'an edge' over its competitors, while someone who trains hard to be the best at what they do could have 'the edge' when it comes to competing. Another way to use this expression is in terms of someone's personality. Some people are described as having 'a bit of an edge' to them if it seems that they might get angry or forceful at a moment's notice.