Edge-connectivity of a weighted undirected graph
This is generally known as "edge connectivity augmentation", and there is a lot of literature on the subject. There are several fast algorithms for the task, and they come in different flavors depending on what you know about your graph, or what you specifically care about.

A basic question is "what is the minimum number of edges (or minimum cost of edges) to be added to the graph $G$ to increase its edge connectivity to a certain value?".