Properties [of network systems]

Random behavior seems to occur in a few out of every thousand randomly selected rules of the kind shown here. In case (c), the following gives a list of the numbers of nodes generated up to step t:

FoldList[Plus, 1, Join[{1, 4, 12, 10, -20, 6, 4}, Map[d, IntegerDigits[Range[4, t - 5], 2]]]]

d[{___, 1}] = 1

d[{1, p : 0 .., 0}] := -Apply[Plus, 4 Range[Length[{p}]] - 1] + 6

d[{__, 1, p : 0 .., 0}] := d[{1, p, 0}] - 7

d[{___, p : 1 .., q : 0 ..., 1, 0}] := 4 Length[{p}] + 3 Length[{q}] + 2

d[{___, p : 1 .., 1, 0}] := 4 Length[{p}] + 2