Abstract: A random recursive tree is a rooted tree constructed by successively choosing a vertex uniformly at random and adding a child to the selected vertex. A random preferential attachment tree is grown in a similar manner, but the vertex selection is made proportional to a linear function of the number of children of a vertex. Preferential attachment trees can be considered as the tree version of the Barabasi-Albert preferential attachment model.

We consider a red-blue coloring of the vertices of preferential attachment trees, which we call a broadcasting-induced coloring: the root is either red or blue with equal probability, while for a fixed value p between 0 and 1, every other vertex is assigned the same color as its parent with probability p and the other color otherwise. In this talk I will discuss properties of preferential attachment trees with broadcasting-induced colorings, including limit laws for the number of vertices, clusters (maximal monochromatic subtrees) and leaves of each color. The main focus of the talk will be on the size of the root cluster, that is, the maximal monochromatic subtree containing the root.

Joint work with Cecilia Holmgren and Stephan Wagner.