I will talk about a few graph-theoretic metrics then introduce the concept of refinements on a class of functions that include all metrics. As a case study, we will construct various refinements on the shortest-path distance. Consequently, we obtain a few “better” versions of the Erdos number. In the course of our investigation, we realized various construction of metrics can be unified under a rather natural concept that we called monotonic monoid norm. This is a joint work with Kayla Lock and Alex Wittmond.