Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors

[Properties of] random networks

The pictures below show networks in which each of a set of n nodes has as its successor a node that is chosen at random from the set. The total number of possible such networks is nn. For large n, the average number of distinct cycles in all such networks is Sqrt[π/2] Log[n], and the average length of these cycles is Sqrt[π/8 n]. The average fraction of nodes that have no predecessor is (1 - 1/n)n or 1/ⅇ in the limit n->∞. Note that processes such as cellular automaton evolution do not yield networks whose properties are particularly close to those of purely random ones.

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