I would probably call \\(\mathbf{Cost^{op}}\\) \\(\mathbf{Gross}\\), or \\(\mathbf{Reward}\\). It's useful when you are interested in the maximum of alternate paths. What's the most you can get out of going from a to b.

Unfortunately, even though this feels like it should work to describe how much energy you can get by running a heat engine, or how much money you can make on a paper route, any non zero loops force the co distance to infinity.

Unfortunately, even though this feels like it should work to describe how much energy you can get by running a heat engine, or how much money you can make on a paper route, any non zero loops force the co distance to infinity.