Two important sensitivity issues over shortest path problems have been discussed. One is the problem of updating shortest paths when nodes are added and when the lengths of some arcs are increased or decreased. The other is the problem of calculating arc tolerances, that is the maximum increase of decrease in the length of a single arc without changing a given optimal tree. In this paper, assuming that there exists a parameter of interest whose perturbation causes the simultaneous changes in arc lengths, we find the invariance condition on these simultaneous changes such that the shortest path between two specified nodes remains unchanged.