Let G be a directed graph on N vertices chosen at random, conditional on the requirement that the out-degree of each vertex is 1 and the in-degree of each vertex is either 0 or 2. The "periodic" points of G are those contained in a cycle. What do we know about the statistics of G? For instance, what is the mean number of periodic points, and how do the cycle lengths look?
By comparison, a random directed graph with all out-degrees 1 (which is to say, the graph of a random function from vertices to vertices) has on order of sqrt(N) periodic points on average.
(Motivation: the graph of a quadratic rational function f acting on P^1(F_q) looks like this, and I'm wondering what the "expected" dynamics are.)