Correlated uniform attachment trees

Abstract

We introduce and study a new model of correlated uniform attachment (UA) trees, where correlation is sprinkled throughout the time evolution of the process. In this model, two UA trees are grown in parallel, and at each time step a new node is added to each tree, with an edge between it and a uniformly chosen existing vertex in the respective tree. The two choices of attachment are correlated: with probability $\alpha$, the edges attach to nodes with the same time label in both trees, and with probability $1-\alpha$, the choices are made independently. We study fundamental detection and estimation questions for this model, given two \emph{unlabeled} trees. In our main result, we construct a consistent estimator of the correlation parameter $\alpha$, as the size of the trees goes to infinity. The construction of our statistic relies on two key ideas. First, we use Jordan centrality to identify subsets of vertices of each tree whose intersection has a sufficient number of common early vertices. The second idea is that, across multiple time scales, it is possible to approximately determine the labels of vertices that have attached to these early vertices, using the sizes of fringe subtrees. Our analysis includes novel quantitative bounds on the fraction of early vertices that remain central, which are of independent interest in the network archaeology literature.