Random position of nodes in hierarchical tree with alternate paths - graphviz

Question

I'm struggling with directed graph hierarchy tree visualization (distributed network in this case). Using graphviz: dot I have this tree:
digraph G {
node[width=0.14, height=0.14];
edge[weight=3, color = "0.000 0.000 0.0"];
1 -> 2;
1 -> 3;
edge[weight=2, color = "0.000 0.000 0.175"];
2 -> 4;
2 -> 5;
3 -> 6;
3 -> 7;
edge[weight=1, color = "0.000 0.000 0.825"];
3 -> 4;
3 -> 5;
2 -> 6;
2 -> 7;
}
Resulting in:
You can see that the primary paths (edges with higher weights) are crossed. The goal is to have the preferred path nodes as close as possible. I can do it by changing the order of nodes randomly so the output looks like this:
However I don't want to think about the ordering algorithm as I want to automate this on hundreds of nodes.
From documentation:
The weight of an edge provides another way to keep edges straight. An edge’s weight suggests some measure of an edge’s importance; thus, the heavier the weight, the closer together its nodes should be. dot causes edges with heavier weights to be drawn shorter and straighter.
But this is not the case as the nodes are drawn in random order. What am I doing wrong?

Answer 1

Probably not the answer you were looking for, but since dot doesn't seem to work with edge weights in this case:
You may just use constraint=false for the lightgray edges.

Answer 2

Could you live with a solution like this, which places invisible ranks and invisible edges to enforce the left-to-right ordering of your nodes? While it adds some white space to the left of the diagram, at least it neatly solves the random-ordering issue within each rank, and lends itself to being automated.
digraph G {
node[width=0.14, height=0.14];
{
rankdir="TB";
edge [style=invis];
rank1 [style=invis];
rank2 [style=invis];
rank3 [style=invis];
rank1 -> rank2 -> rank3 [style=invis];
}
edge[weight=3, color = "0.000 0.000 0.0"];
1 -> 2;
1 -> 3;
edge[weight=2, color = "0.000 0.000 0.175"];
2 -> 4;
2 -> 5;
3 -> 6;
3 -> 7;
{
edge [style=invis];
rank=same;
rank2 -> 2 -> 3;
}
edge[weight=1, color = "0.000 0.000 0.825"];
3 -> 4;
3 -> 5;
2 -> 6;
2 -> 7;
{
rank=same;
edge [style=invis];
rank3 -> 4 -> 5 -> 6 -> 7 ;
}
}