Hello Friend! This is my first "help" post here on math help forum so I'm gonna try my best!
The biggest problem with negative weights is cycles. I'm not really sure how/if to upload/insert images so I hope my description will be good enough .
Lets say you have three nodes connected in a graph. If positive weights represent "distance", The negative weights represent a "reward". Given the negative weights, you can get locked into a "reward" cycle where you can just keep moving around in a circle to get reward. Three nodes S, T, and U (no F, har har, little joke) are connected by edges:
S-T = -1
T-U = 2
S-U = 2
S-T-U is a minimum cost path (-1+2 = 1). How, we'll add a constant (2) to make all the weights positive:
S-T = 1
T-U = 4
S-U = 4
S-T-U is no longer a minium cost path (1 + 4 = 5 > (S-U) 4). When we return the shortest path (S-U), and subtract our added value (2) we'll get 2, but the answer is obviously bogus since S-U is NOT a shortest path answer in the original graph.
There's the Bellmen-Ford algorithm for negative weights. It will report a shortest path given negative cost weights, or report the existance of a negative cost cycle. Take a look on google if you interest!
Have a good day