Geodesic
Layers of the graph of $k$-fold iteration of the uniform random mapping
Matematičeskie voprosy kriptografii, Tome 10 (2019) no. 1, pp. 73-82 Cet article a éte moissonné depuis la source Math-Net.Ru

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The probabilistic characteristics of the graph of $k$-fold iteration of uniform random mapping are studied. Exact expressions and estimates for the distribution of the distances from the vertices to cycles are obtained. Formulas for expected values of the numbers of vertices on the given distance from cycles and for the distribution function of the height of a random vertex are derived. 
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V. O. Mironkin. Layers of the graph of $k$-fold iteration of the uniform random mapping. Matematičeskie voprosy kriptografii, Tome 10 (2019) no. 1, pp. 73-82. http://geodesic.mathdoc.fr/item/MVK_2019_10_1_a4/

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