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

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

** Next:** Reducing the problem to
** Up:** Small world problem estimated
** Previous:** Stating the problem
** Contents**
Tiziano Mengotti
2004-03-27