We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree distributions and, more generally, the limiting distribution of subgraphs in balls of any given radius $k$ around a random vertex in the preferential attachment graph. We also establish the finite-volume corrections which give the approach to the limit.
"Asymptotic behavior and distributional limits of preferential attachment graphs." Ann. Probab. 42 (1) 1 - 40, January 2014. https://doi.org/10.1214/12-AOP755