The friendship paradox for trees

TL;DR

This paper investigates the friendship paradox on trees, analyzing the distribution of vertices with positive, neutral, or negative friendship bias in finite and infinite trees, revealing structural properties and correlations based on offspring distribution.

Contribution

It introduces the concept of significance in finite trees, providing bounds, and analyzes the densities and correlations of vertex types in infinite Galton-Watson trees, extending understanding of the friendship paradox.

Findings

Number of positive vertices is at least as large as negative in finite trees.

Densities of vertex types vary in infinite trees depending on offspring distribution.

Edge type correlations depend on offspring distribution conditions.

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 positively or negatively correlated.