We consider the preferential attachment model. This is a growing random graphsuch that at each step a new vertex is added and forms $m$ connections. Theneighbors of the new vertex are chosen at random with probability proportionalto their degree. It is well known that the proportion of nodes with a givendegree at step $n$ converges to a constant as $nightarrowinfty$. The goal ofthis paper is to investigate the asymptotic distribution of the fluctuationsaround this limiting value. We prove a central limit theorem for the jointdistribution of all degree counts. In particular, we give an explicitexpression for the asymptotic covariance. This expression is rather complex, sowe compute it numerically for various parameter choices. We also use numericalsimulations to argue that the convergence is quite fast. The proof relies onthe careful construction of an appropriate martingale.
Asymptotic normality of degree counts in a general preferential attachment model
Simone Baldassarri
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2022-01-01
Abstract
We consider the preferential attachment model. This is a growing random graphsuch that at each step a new vertex is added and forms $m$ connections. Theneighbors of the new vertex are chosen at random with probability proportionalto their degree. It is well known that the proportion of nodes with a givendegree at step $n$ converges to a constant as $nightarrowinfty$. The goal ofthis paper is to investigate the asymptotic distribution of the fluctuationsaround this limiting value. We prove a central limit theorem for the jointdistribution of all degree counts. In particular, we give an explicitexpression for the asymptotic covariance. This expression is rather complex, sowe compute it numerically for various parameter choices. We also use numericalsimulations to argue that the convergence is quite fast. The proof relies onthe careful construction of an appropriate martingale.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
