Geodesic
Sizes of Trees in a Random Forest and Configuration Graphs
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 298-315.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider Galton–Watson random forests with $N$ rooted trees and $n$ nonroot vertices. The distribution of the number of offspring of the critical homogeneous branching process generating a forest has infinite variance. Such branching processes are used in the study of the structure of random configuration graphs designed for simulating complex communication networks. We prove theorems on the limit distributions of the number of trees of a given size for various relations between $N$ and $n$ as they tend to infinity. 
@article{TM_2022_316_a19,
     author = {Yu. L. Pavlov and I. A. Cheplyukova},
     title = {Sizes of {Trees} in a {Random} {Forest} and {Configuration} {Graphs}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {298--315},
     publisher = {mathdoc},
     volume = {316},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_316_a19/}
}
TY  - JOUR
AU  - Yu. L. Pavlov
AU  - I. A. Cheplyukova
TI  - Sizes of Trees in a Random Forest and Configuration Graphs
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2022
SP  - 298
EP  - 315
VL  - 316
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2022_316_a19/
LA  - ru
ID  - TM_2022_316_a19
ER  - 
%0 Journal Article
%A Yu. L. Pavlov
%A I. A. Cheplyukova
%T Sizes of Trees in a Random Forest and Configuration Graphs
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2022
%P 298-315
%V 316
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2022_316_a19/
%G ru
%F TM_2022_316_a19
Yu. L. Pavlov; I. A. Cheplyukova. Sizes of Trees in a Random Forest and Configuration Graphs. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 298-315. http://geodesic.mathdoc.fr/item/TM_2022_316_a19/

[1] Bollobas B., “A probabilistic proof of an asymptotic formula for the number of labelled regular graphs”, Eur. J. Comb., 1:4 (1980), 311–316 | DOI | MR | Zbl

[2] Durrett R., Random graph dynamics, Cambridge Ser. Stat. Probab. Math., 20, Cambridge Univ. Press, Cambridge, 2007 | MR | Zbl

[3] Dwass M., “The total progeny in a branching process and a related random walk”, J. Appl. Probab., 6:3 (1969), 682–686 | DOI | MR | Zbl

[4] W. Feller, An Introduction to Probability Theory and Its Applications, v. 2, J. Wiley and Sons, New York, 1971 | MR | MR | Zbl

[5] Van der Hofstad R., Random graphs and complex networks, v. 1, Cambridge Ser. Stat. Probab. Math., 43, Cambridge Univ. Press, Cambridge, 2017 | MR | Zbl

[6] Van der Hofstad R., Random graphs and complex networks, v. 2. To appear, Cambridge Univ. Press., Cambridge | Zbl

[7] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publ., Groningen, 1971 | MR | Zbl

[8] N. I. Kazimirov and Yu. L. Pavlov, “A remark on the Galton–Watson forests”, Discrete Math. Appl., 10:1 (2000), 49–62 | DOI | MR | Zbl

[9] V. F. Kolchin, Random Mappings, Optim. Softw., New York, 1986 | MR | Zbl

[10] V. F. Kolchin, Random Graphs, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[11] A. B. Mukhin, “Local limit theorems for lattice random variables”, Theory Probab. Appl., 36:4 (1992), 698–713 | DOI | MR

[12] Yu. L. Pavlov, “Limit distributions of the number of trees of a given size in a random forest”, Discrete Math. Appl., 6:2 (1996), 117–133 | DOI | MR | Zbl

[13] Pavlov Yu.L., Random forests, VSP, Utrecht, 2000 | MR

[14] Yu. L. Pavlov, “The maximum tree of a random forest in the configuration graph”, Sb. Math., 212:9 (2021), 1329–1346 | DOI | MR | Zbl