Chapter 9: Fundamental Physics

Section 7: Space as a Network

Finding layouts [for networks]

One way to lay out a network g so that network distances in it come as close as possible to ordinary distances in d-dimensional space, is just to search for values of the x[i, k] which minimize a quantity such as

With[{n = Length[g]}, Apply[Plus, Flatten[(Table[Distance[g, {i, j}], {i, n}, {j, n}]2 - Table[Sum[(x[i, k] - x[j, k])2, {k, d}], {i, n}, {j, n}])2]]]

using for example FindMinimum starting say with x[1, _] 0 and all the other x[_, _] Random[]. Rarely is there a unique minimum that can be found, but the approach nevertheless seems to work fairly well whenever a good layout exists in a particular number of dimensions. One can imagine weighting different network distances differently, but usually I have found that equal weightings work best. If one ignores all constraints beyond network distance 1, then one is in effect just trying to build the network out of identical rigid rods. It turns out that this is almost always possible even in 2D (though not in 1D); the only exception is the tetrahedron network. And in fact very few trivalent structures are rigid, in the sense the angles between rods are uniquely determined. (In 3D, for example, this is true only for the tetrahedron.)

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From Stephen Wolfram: A New Kind of Science [citation]