Random Friend Trees

• Authors: Louigi Addario-Berry, Simon Briend, Luc P Devroye, Serte Donderwinkel, Céline Kerriou and Gábor Lugosi
• Vol. 69, No. 1, Article e70082, August 2026

We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saramäki and Kaski (2004). Vertices are attached sequentially, one by one, by selecting an existing target vertex and connecting to one of its neighbours (or friends), chosen uniformly at random. This model has interesting emergent properties, such as a highly skewed degree sequence. In contrast to the preferential attachment model, these emergent phenomena stem from a local rather than a global attachment mechanism. The structure of the resulting tree is also strikingly different from both the preferential attachment tree and the uniform random recursive tree: every edge is incident to a macro-hub of asymptotically linear degree, and with high probability all but at most (Formula presented.) vertices in a tree of size (Formula presented.) are leaves. We prove various results on the neighbourhood of fixed vertices and edges, and we study macroscopic properties such as the diameter and the degree distribution, providing insights into the overall structure of the tree. We also present a number of open questions on this model and related models.