I have studied the landmark papers of Rubin et al, Harris, Flajolet on Random mapping statistics. I have also read some follow up papers. They provided analysis of the structure of random maps. My question is whether there is any literature where multiple random maps are unioned together.
For example suppose there are two distinct random functions f_1, f_2 : N--->N. Then consider the union of the two random maps generated over the set of vertices N. Then what can we say about the distributions on the number of connected components, component size, number of cycles, six-length, cycle-length etc. in this union graph?
I will really appreciate if anyone can lead me towards any good reference.
A closely related concept is $k$-out graphs. These are directed graphs generated in nearly the same way, except that for every vertex, exactly $k$ edges from that vertex are chosen uniformly at random. (The underlying undirected graph is also often studied.)
In a union of random bijections $f_1 \cup f_2 \cup \dots \cup f_k$, also viewed as a directed graph, not every vertex is guaranteed to have out-degree $k$; we may have $f_i(x) = f_j(x)$ for some $i \ne j$, in which case the out-degree of $x$ is reduced. Asymptotically, for constant or even slowly growing $k$, this should not happen too often. For example, for $k=2$, the probability is $\frac1n$ that a particular vertex will have out-degree $1$, so the number of such vertices is $\text{Binomial}(n, \frac1n)$, which approaches $\text{Poisson}(1)$ in the limit; in expectation, there is only one such vertex, and with probability tending to $e^{-1}$, there are no such vertices.
This means that the union of random bijections will have many of the same properties as the corresponding $k$-out graph, but maybe not all of them. (For example, it is not guaranteed to have the same minimum degree and therefore not the same connectivity.)
Frieze and Karoński's Introduction to Random Graphs has a chapter on $k$-out graphs which can be a starting point. (It also has some citations scattered throughout that chapter.)