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.
