The friendship paradox for trees

Electronic Journal of Probability, vol. 31, pp. 1–26

Abstract

We analyse the friendship paradox on finite and infinite trees. In particular, we monitor the vertices for which the friendship-bias is positive, neutral and negative, respectively. For an arbitrary finite tree, we show that the number of positive vertices is at least as large as the number of negative vertices, a property we refer to as significance, and derive a lower bound in terms of the branching points in the tree. For an infinite Galton-Watson tree, we compute the densities of the positive and the negative vertices and show that either may dominate the other, depending on the offspring distribution. We also compute the densities of the edges having two given types of vertices at their ends, and give conditions in terms of the offspring distribution under which these types are negatively correlated.

Authors 4

1. Rajat Subhra Hazra Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands

2. Frank den Hollander Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands

3. Nelly Litvak

   Eindhoven University of Technology

   Affiliation as printed

   Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

4. Azadeh Parvaneh Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands