Analogic machine to compute solution

Imagine a table with a random graph on it. Unlike normal graph theory, we force all edges to the same length. The reason for this will soon be clear; we take the node that represents ourself and elevate it over the table, the rest of the graph falls down in disorderly fashion. Equipped with scissors, we cut edges that go back to upper levels of the graph, until our graph turns into a tree; as soon as all double edges are cut, the entire tree falls down like a line thanks to the force of gravity. Levels in the tree are all of the same length, thanks to our initial constraint.

We take the tree and put it level by level into a bin. We count how many levels we put into the bin until we reach the presidential node; this is then the path length between us and the President. Unfortunately, depending on how we cut the graph, the path length might change a lot.

Notice that because we force the same edge length on all edges, a random graph might not fit on the table, but partially elevate into air. And even with 3 dimensions, there will be random graphs that we cannot construct, if they have lot of edges and if these edges cannot stretch. Nonetheless, these graphs exist in high-dimensional spaces we cannot easily imagine, with $d>3$.

This is strong evidence for a dimensional path to the solution.

Figure 9: Forcing fixed length edges on a simplified one dimensional table.