Persistent Hubs and Sharp Asymptotics in Eigenvector Preferential Attachment Trees

Source abstract

We prove sharp almost-sure asymptotics for eigenvector preferential attachment trees in which each attachment uses eigenvector centrality with probability $α\in(0,1]$ and uniform selection with probability $1-α$. A unique persistent hub eventually has the maximum degree, which is asymptotic to $γ^2t^{2/3}$, where $γ=(9α/10)^{1/3}$. The Perron-Frobenius eigenvalue is asymptotic to $γt^{1/3}$, and the second-largest degree is asymptotic to $2γt^{1/3}$. The hub and its neighbors each carry asymptotically half of the squared Perron mass. We also identify the centrality profile by degree and an explicit mixture of geometric laws for the degree of a uniformly sampled vertex. A limiting distance profile quantifies the dandelion geometry, and vertices beyond the hub's neighbors have maximum degree of order at most $\log t$.