karger's algorithm contracting nodes not edges Karger's algorithm works by contracting edges, not merging nodes (this is different because nodes need not share an edge).

Is there a reason why this is so?
The run-time of Karger-Stein is usually represented as a function of the number of vertices $n$, not of the number of edges. Therefore, placing an edge of weight $0$ between two vertices that previously did not share an edge would make your algorithm identical to theirs and not change the run-time.