>There's nothing about the algorithm that says that distance must be the sum of all the edge weights along a path
I think it relies on the assumption that d(a, c) >= d(a, b) + d(b, c). You could relax this of course, but then you're just back to bellman-ford algorithm.
When your distance aggregation is instead multiplicative and you allow multipliers < 1, this no longer holds.
Edit: I see your footnote already mentions this. I agree, the key point here isn't really whether things add or not, it's whether your aggregate distances are monotonically increasing or not. If they are, you can use dijkstra. If not, you have to use bellman ford or the fancier algorithms like the one in the arxiv.
You're right about the difference between Dijkstra and Bellman-Ford. But I thought the question was "why would you want negative edge weights?", and this doesn't seem like an answer to that.
I think it relies on the assumption that d(a, c) >= d(a, b) + d(b, c). You could relax this of course, but then you're just back to bellman-ford algorithm.
When your distance aggregation is instead multiplicative and you allow multipliers < 1, this no longer holds.
Edit: I see your footnote already mentions this. I agree, the key point here isn't really whether things add or not, it's whether your aggregate distances are monotonically increasing or not. If they are, you can use dijkstra. If not, you have to use bellman ford or the fancier algorithms like the one in the arxiv.