Interesting issue, thanks for posting about that. I attach a modified version of your gexf: I added a 3rd node and a 2nd edge. This second edge has decreasing weights with time, while the 1st edge in your original file had increasing weight with time. The value of the weight for the 2 edges crosses (is equal) when the date is 2020.
Before 2020: 1st edge is thin, 2nd edge is thick. No evolution until 2020.
At 2020: both edges equally thin (as said above they have the same weight value at this date)
After 2020: 1st edge is thick, 2nd edge is thin. No evolution until the end of the timeline.This suggests that the thickness of edges is established relatively between edges
-> With just one edge on the graph: no comparison to be made. Even if the weight of this edge varies, its thickness will remain constant because there is no other edge to compare it to.
-> several edges on the graph: at each point in time, Gephi checks all weights of all edges at this point in time
. It establishes a range of thickness function of the lowest to the highest weight values at this point in time.
You can have a finer grained view on the evolution of the values of the weight of individual edges by using a dynamic ranking on edge's weight (in the ranking panel, select edges, click on the tiny icon just left to "Auto Apply", then click on Auto apply. Then run the timeline.) You should see the edge' color vary according to weight's values.
This applies to the edge' color or label size though, not edge thickness. Here, even with just one edge on your graph, you'd see its color change with time.
My view? This could definitely be improved, maybe by simply treating the edge thickness as any other visual attribute (like edge color or label size, which can be controlled from the ranking panel). I open an issue on Github:https://github.com/gephi/gephi/issues/836