The friendship paradox for trees

TL;DR

This work quantifies the friendship paradox on tree structures, introducing vertex types by the sign of the local bias and distinguishing when the paradox is significant. It provides a complete finite-tree result with an explicit lower bound tied to branching points, and a detailed infinite-tree analysis for Galton–Watson limits, deriving explicit densities f^χ of vertex types and f^{\tilde{χ}χ} of edge types. The densities are expressed via size-biased offspring distributions and sums of i.i.d. offspring variables, revealing that significance need not hold for all infinite trees and that positive/negative vertex types exhibit nontrivial correlations along edges. The results yield a rigorous framework for understanding the local structure of the friendship paradox in sparse, tree-like networks and pose questions about percolation and extensions to other random-tree models. The formulas enable precise numerical assessment of the paradox in a broad class of random trees and illuminate how branching, degree distributions, and local geometry shape the phenomenon.

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.

Figures (4)

• Figure 1: Four realisations of generations $0,\ldots,6$ of an infinite rooted Galton-Watson tree with an offspring distribution that is Poisson$(\lambda)$ with $\lambda = 0.1, 0.5, 1, 2$, respectively. Indicated are the locations of the positive (= red), neutral (= circle) and negative (= blue) vertices in the infinite tree. Negative vertices have a tendency to be adjacent to positive vertices. Smaller values of $\lambda$ tend to produce a higher proportion of neutral vertices, while larger values of $\lambda$ tend to produce a higher proportion of positive vertices. In addition, for small values of $\lambda$ the root is almost always positive or neutral, while for large values of $\lambda$ the probability of the root being negative increases, although it remains smaller than the probability of the root being positive or neutral. This is consistent with the findings on the friendship paradox in sparse Erdős-Rényi random graphs reported in HHP1. Indeed, the local limit of the Erdős-Rényi random graph with edge density $\lambda/n$ is the rooted Galton-Watson tree with offspring distribution Poisson$(\lambda)$ (see vdH2).
• Figure 2: Plots of $k \mapsto f(\tilde{k}, k, \lambda) = \mathbb{P}(\tilde{k} + S_k - k(k+1) > 0)$ for $\tilde{k} \in \{1,2,5,8,10\}$ and $\lambda \in\{1.5,2,3,7\}$. The function is non-increasing in $k$ for all displayed values, except for $\lambda=7$ and $\tilde{k}=1$, where a slight increase is observed from $k=1$ to $k=2$.
• Figure 3: Indicated in black is the subtree $\mathcal{T}_{n_2}$ of the full tree $\mathcal{T}_n$.
• Figure 4: Indicated in black is the subtree $\mathcal{T}_{n_3}$ of the full tree $\mathcal{T}_n$.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Example 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Example 1.9
• Lemma 2.1