The root cluster after percolation on preferential attachment trees

Abstract

The class of linear preferential attachment trees includes recursive trees, plane-oriented recursive trees, binary search trees, and increasing $d$-ary trees. Bond percolation with parameter $p$ is performed by considering every edge in a graph independently, and either keeping the edge with probability $p$ or removing it otherwise. The resulting connected components are called clusters. In this extended abstract, we demonstrate how to use methods from analytic combinatorics to compute limiting distributions, after rescaling, for the size of the cluster containing the root. These results are part of a larger work on broadcasting induced colorings of preferential attachment trees.