期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2018
卷号：XV
期号：2
页码：1431-1445
DOI：10.30757/ALEA.v15-53
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We consider the connected component of the partial duplication modelfor a random graph, a model which was introduced by Bhan, Galas and Dewey as amodel for gene expression networks. The most rigorous results are due to Hermannand Pfa elhuber (2016), who show a phase transition between a subcritical casewhere in the limit almost all vertices are isolated and a supercritical case where theproportion of the vertices which are connected is bounded away from zero.We study the connected component in the subcritical case, and show that, whenthe duplication parameter p < e1, the degree distribution of the connected componenthas a limit, which we can describe in terms of the stationary distributionof a certain Markov chain and which follows an approximately power law tail, withthe power law index predicted by Ispolatov et al. (2005). Our methods involveanalysing the quasi-stationary distribution of a certain continuous time Markovchain associated with the evolution of the graph.
