The multi-level friendship paradox for sparse random graphs

TL;DR

This paper extends the analysis of the friendship paradox to multi-level biases in sparse random graphs, exploring the limits of empirical distributions under different exploration methods and their asymptotic behaviors.

Contribution

It introduces the concept of multi-level friendship biases and characterizes their limiting distributions in sparse random graphs, including the effects of exploration methods and limit interactions.

Findings

Limits of empirical distributions identified for non-backtracking exploration.

Limits commute for a broad class of graphs including Galton-Watson trees.

Conditions under which limits do not commute are also characterized.

Abstract

In Hazra, den Hollander and Parvaneh (2025) we analysed the friendship paradox for sparse random graphs. For four classes of random graphs we characterised the empirical distribution of the friendship biases between vertices and their neighbours at distance $1$, proving convergence as $n\to\infty$ to a limiting distribution, with $n$ the number of vertices, and identifying moments and tail exponents of the limiting distribution. In the present paper we look at the multi-level friendship bias between vertices and their neighbours at distance $k \in \mathbb{N}$ obtained via a $k$-step exploration according to a backtracking or a non-backtracking random walk. We identify the limit of empirical distribution of the multi-level friendship biases as $n\to\infty$ and/or $k\to\infty$. We show that for non-backtracking exploration the two limits commute for a large class of sparse random graphs, including those that locally converge to a rooted Galton-Watson tree. In particular, we show that the same limit arises when $k$ depends on $n$, i.e., $k=k_n$, provided $\lim_{n\to\infty} k_n = \infty$ under some mild conditions. We exhibit cases where the two limits do not commute and show the relevance of the mixing time of the exploration.