On définit de nouveaux processus de naissance à temps discret; la population est, à chaque instant, organisée en graphe. Pour obtenir la -ième génération on remplace aléatoirement les sommets de la -ième génération par des graphes que l’on accroche convenablement les uns aux autres. On autorise une certaine dépendance entre les substitutions de sommets voisins. On étudie, pour certains processus surcritiques, la croissance de la population et la structure des graphes générés : sous des hypothèses convenables la fréquence d’apparition d’un sous-graphe déterminé dans la -ième génération tend, presque sûrement, vers une constante. Ces processus rendent compte de constructions de B. Mandelbrot.
New birth processes are defined. Population at time is endowed with a graph structure. To obtain the -th generation, the vertices of the -th generation are replaced at random by graphs suitably bound together. Replacements of neighboring vertices may be statistically dependent. The growth of the population is studied and so is the structure of generated graphs: under suitable assumptions, the frequency of appearance of a fixed pattern in the -th generation tends to a constant with probability one. These processes originate from the work of B. Mandelbrot.
@article{AIF_1981__31_4_187_0,
author = {Peyri\`ere, Jacques},
title = {Processus de naissance avec interaction des voisins, \'evolution de graphes},
journal = {Annales de l'Institut Fourier},
volume = {31},
year = {1981},
pages = {187-218},
doi = {10.5802/aif.855},
mrnumber = {84d:60126},
zbl = {0452.60089},
language = {fr},
url = {http://dml.mathdoc.fr/item/AIF_1981__31_4_187_0}
}
Peyrière, Jacques. Processus de naissance avec interaction des voisins, évolution de graphes. Annales de l'Institut Fourier, Tome 31 (1981) pp. 187-218. doi : 10.5802/aif.855. http://gdmltest.u-ga.fr/item/AIF_1981__31_4_187_0/
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