On the influence of the seed graph in the preferential attachment model

We are interested in the following question: suppose we generate a large graph according to the linear preferential attachment model—can we say anything about the initial (seed) graph? A precise answer to this question could lead to new insights for the diverse applications of the preferential attachment model. In this work we focus on the case of trees grown according to the preferential attachment model. We first show that the seed has no effect from a weak local limit point of view. On the other hand, we conjecture that different seeds lead to different distributions of limiting trees from a total variation point of view. We take some steps in proving this conjecture by focusing on star seeds.