The number of descendants in a random directed acyclic graph
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| Titel: | The number of descendants in a random directed acyclic graph |
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| Autoren: | Janson, Svante, 1955 |
| Quelle: | Random structures & algorithms (Print). 64(3):768-803 |
| Schlagwörter: | random acyclic digraph, random circuit, Yule tree |
| Beschreibung: | We consider a well-known model of random directed acyclic graphs of order n$$ n $$, obtained by recursively adding vertices, where each new vertex has a fixed outdegree d > 2$$ d\geqslant 2 $$ and the endpoints of the d$$ d $$ edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number X(n)$$ {X}<^>{(n)} $$of vertices that are descendants of n$$ n $$. We show that X(n)/n(d-1)/d$$ {X}<^>{(n)}/{n}<^>{\left(d-1\right)/d} $$ converges in distribution; the limit distribution is, up to a constant factor, given by the d$$ d $$th root of a Gamma distributed variable with distribution Gamma(d/(d-1))$$ \Gamma \left(d/\left(d-1\right)\right) $$. When d=2$$ d=2 $$, the limit distribution can also be described as a chi distribution chi(4)$$ \chi (4) $$. We also show convergence of moments, and find thus the asymptotics of the mean and higher moments. |
| Dateibeschreibung: | electronic |
| Zugangs-URL: | https://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-528253 https://doi.org/10.1002/rsa.21195 |
| Datenbank: | SwePub |
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Nájsť tento článok vo Web of Science | Abstract: | We consider a well-known model of random directed acyclic graphs of order n$$ n $$, obtained by recursively adding vertices, where each new vertex has a fixed outdegree d > 2$$ d\geqslant 2 $$ and the endpoints of the d$$ d $$ edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number X(n)$$ {X}<^>{(n)} $$of vertices that are descendants of n$$ n $$. We show that X(n)/n(d-1)/d$$ {X}<^>{(n)}/{n}<^>{\left(d-1\right)/d} $$ converges in distribution; the limit distribution is, up to a constant factor, given by the d$$ d $$th root of a Gamma distributed variable with distribution Gamma(d/(d-1))$$ \Gamma \left(d/\left(d-1\right)\right) $$. When d=2$$ d=2 $$, the limit distribution can also be described as a chi distribution chi(4)$$ \chi (4) $$. We also show convergence of moments, and find thus the asymptotics of the mean and higher moments. |
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| ISSN: | 10429832 10982418 |
| DOI: | 10.1002/rsa.21195 |