The answer is in the question’s title:
Graph (network) vis ualization tool
You want GraphViz.
I have been using it for decades now, and can highly recommend it. It is stable, well documented, and well supported (including 7,277 questions & answers on Stack Overflow).
It takes plain text as input and generates diagrams, so your
Paris, Oslo, 0.25
Paris, New-York, 0.0001
New-York, Wasington, 0.1
could be easily manipulated to become input.
Perhaps the S.O question Graphviz, changing the size of edge will be poof help?
There is official GraphViz documentation for Node, Edge and Graph Attributes.
The GraphViz Pocket Reference gives
• [label="Some Label"]Labels the Edge (Useful for Weights)
• [color="red"]Colors the Vertex (Useful for Paths)
• [penwidth=2.0]Adjusts the thickness of the edge line, Very useful for Paths
I will confess that I am not a “graph person”, and use it only for diagramming, but here is enough reference to graphs and GraphViz online that I am confident that it is what you are looking for. Please let me know whether I am right or wrong :-)
a clear into / tutorial, including weighting