[Generating] random networks

One way to generate the connections for a "completely random" trivalent network with n nodes is just to apply a random permutation:

RandomNetwork[n_?EvenQ] := Partition[Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Floor[Range[1, n + 2/3, 1/3]]], 2]

Networks obtained in this way are usually connected, but will almost always contain self-loops and multiple edges. Properties of random networks are discussed on page 963. A convenient way to get somewhat random planar networks is from 2D Voronoi diagrams of the kind discussed on page 987.