In a 2-parameter scale free model of random graphs it is shown that the asymptotic degree distribution is the same in the neighbourhood of every vertex. This degree distribution is still a power law with characteristic exponent 2, but this exponent is different from the one observed in the whole graph.
"Degree distribution nearby the origin of a preferential attachment graph." Electron. Commun. Probab. 12 276 - 282, 2007. https://doi.org/10.1214/ECP.v12-1299