I don't believe that's possible in the general case. You can easily imagine a graph where your source node has only a single connection to the rest of the graph. If that edge changes, the weight of every path from the source will change (although not the paths themselves).
There are other pathological cases you can imagine where even a single edge change would dramatically affect pathing. Imagine a graph with two almost equally weighted chokepoints that almost all paths going to. A slight change in either chokepoint's weight could cause a majority of the paths in the graph to reroute.
If you're willing to settle for an imperfect solution, one option is to have your game constantly recalculating Dijkstra's algorithm incrementally on the graph. So, in addition to your other game processes, you have another loop that works its way through Dijkstra a few nodes every frame, restarting as soon as its done. You can then tune that to spend as much CPU on it per frame as you can afford. You'll get the benefit of running a full recalculation without having the stop the world, the only problem is that you'll have stale paths for a while.