[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 n^{n}. 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.