Geodesic
The cycle structure of a random nonhomogeneous hypergraph on the subcritical stage of evolution
Diskretnaya Matematika, Tome 19 (2007) no. 4, pp. 52-69
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a random nonhomogeneous hypergraph on $n$ vertices with $M=M(n)$ edges, $M_i=M_i(n)$ edges consist of $i$ vertices, \begin{gather*} \lim_{n\to\infty}M_i/M=c_i,\quad c_i\ge0,\quad i=0,1,\dots,m,\\ c_0+c_1+\dots+c_m=1,\quad M=M_0+M_1+\dots+M_m. \end{gather*} For each edge, vertices are chosen by random and equiprobable sampling with replacement out of $n$ vertices. Under the condition that $n\to\infty$ and $$ 0\lim_{n\to\infty}\frac Mn\Biggl(\sum_{i=2}^mc_ii(i-1)\Biggr)^{-1} $$ we show that the probability that the random hypergraph consists of hypertrees and components with one cycle tends to one. Similar results for random graphs and random homogeneous hypergraphs have been obtained earlier. 
@article{DM_2007_19_4_a3,
     author = {A. V. Shapovalov},
     title = {The cycle structure of a~random nonhomogeneous hypergraph on the subcritical stage of evolution},
     journal = {Diskretnaya Matematika},
     pages = {52--69},
     year = {2007},
     volume = {19},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2007_19_4_a3/}
}
TY  - JOUR
AU  - A. V. Shapovalov
TI  - The cycle structure of a random nonhomogeneous hypergraph on the subcritical stage of evolution
JO  - Diskretnaya Matematika
PY  - 2007
SP  - 52
EP  - 69
VL  - 19
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/DM_2007_19_4_a3/
LA  - ru
ID  - DM_2007_19_4_a3
ER  - 
%0 Journal Article
%A A. V. Shapovalov
%T The cycle structure of a random nonhomogeneous hypergraph on the subcritical stage of evolution
%J Diskretnaya Matematika
%D 2007
%P 52-69
%V 19
%N 4
%U http://geodesic.mathdoc.fr/item/DM_2007_19_4_a3/
%G ru
%F DM_2007_19_4_a3
A. V. Shapovalov. The cycle structure of a random nonhomogeneous hypergraph on the subcritical stage of evolution. Diskretnaya Matematika, Tome 19 (2007) no. 4, pp. 52-69. http://geodesic.mathdoc.fr/item/DM_2007_19_4_a3/

[1] Vantsyan A. G., “Evolyutsiya sluchainykh odnorodnykh gipergrafov”, Veroyatnostnye zadachi diskretnoi matematiki, MIEM, Moskva, 1987, 126–131 | MR

[2] Zykov A. A., “Gipergrafy”, Uspekhi matem. nauk, 29:6 (1974), 89–154 | MR | Zbl

[3] Kolchin V. F., Sistemy sluchainykh uravnenii, MIEM, Moskva, 1988

[4] Kolchin V. F., Sluchainye grafy, Fizmatlit, Moskva, 2004

[5] Stepanov V. E., “Sluchainye grafy”, Voprosy kibernetiki, MGU, Moskva, 1973, 164–185

[6] Shapovalov A. V., “O svyaznosti i porogovykh funktsiyakh podgrafov sluchainykh odnorodnykh gipergrafov”, Diskretnaya matematika, 5:3 (1993), 105–115 | MR | Zbl

[7] Shapovalov A. V., “O chisle strogo sbalansirovannykh podgrafov sluchainykh odnorodnykh gipergrafov”, Diskretnaya matematika, 5:4 (1993), 133–144 | MR | Zbl

[8] Shapovalov A. V., “Raspredeleniya chisel konechnykh podgrafov v sluchainykh neodnorodnykh gipergrafakh”, Diskretnaya matematika, 18:3 (2006), 102–114 | MR | Zbl

[9] Bollobás B., Random graphs, Academic Press, London, 1985 | MR | Zbl

[10] Erdős P., Rényi A., “On the evolution of random graphs”, Publ. Math. Inst. Hung. Acad. Sci., 5 (1960), 17–61 | MR | Zbl