I still don't get it. So you make this graph, with a random number on each edge. Then you duplicate that graph, stack them like a bunkbed, and connect every lower vertex to the one above it. Then you delete all edges with a number below some threshold. Wouldn't the upper and lower graphs still be identical? Since you had the same edge weights in both? So the route from any upper node X to any upper node Y should be 1 less than the route length between X and the point beneath Y.
What did I miss?