Tail index estimation in poisson growing preferential attachment networks

The paper is devoted to the estimation of the tail index in random networks using the Hill estimator applied to the in-degree and out-degree data. The Hill estimator is consistent and asymptotically normal with independent identically distributed (iid) data. Despite the network data being not iid, the Hill estimator can be applied in practice. The minimum distance selection procedure (MDSP) based on Kolmogorov-Smirnov (KS) statistic is widely used to find the required number $k$ of the largest order statistics to calculate the Hill estimate. However, the MDSP may lead to too small values of $k$, and therefore to a large variance of the Hill estimate. In the paper, we propose to use the MDSP based on the Cram\'{e}r-von Mises-Smirnov (CMS) $\omega^2$ statistic. Random networks evolved by a Poisson growing preferential attachment model proposed by Wand and Resnick (2023) are studied. The latter model leads to power-law distributed in- and out-degrees of nodes that approximate better web-graphs. MDSPs with KS and CMS statistics are compared by simulation.