A matrix S = (sij) ∈ R n×n is said to determine a transitional measure for a digraph Γ on n vertices if for all i, j, k ∈ {1, . . . , n}, the transition inequality sij sjk ≤ sik sjj holds and reduces to the equality (called the graph bottleneck identity) if and only if every path in Γ from i to k contains j. We show that every positive transitional measure produces a distance by means of a loga...