Network Topology versus Network Geometry

There is a terminological issue in the study of networks about which I need to rant.

Many people write about network topology, and this is good terminology when referring to connectivity alone. Ignore whether edges have any weights and just consider which ones are connected to which others (with either directed edges or undirected edges). Such terminology is a good extension of the mathematical notion of topology, which only cares about how things are connected to each other but doesn't care a whit about actually measuring distances. Things either are connected or they're not.

The problem is that numerous papers still use the word "topology" when they are talking about the weights of edges. That is a notion of distance, so using the word topology is in fact dead wrong. When one is describing edges weights and playing around with them, the term that people should be using is in fact network geometry, as the point of the mathematical notion of geometry is measuring distance.

So please, please, please distinguish between "network topology" and "network geometry" when you give talks, write papers, etc. Otherwise you are committing a rather egregious mathematical sin.