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The multi-level friendship paradox for sparse random graphs

Rajat Subhra Hazra, Frank den Hollander, Azadeh Parvaneh

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.

The multi-level friendship paradox for sparse random graphs

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 , proving convergence as to a limiting distribution, with 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 obtained via a -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 and/or . 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 depends on , i.e., , provided 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.

Paper Structure

This paper contains 28 sections, 11 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction and outline
  2. The friendship paradox
  3. The multi-level friendship paradox
  4. Outline
  5. Graphs and random walks on random graphs
  6. Graphs
  7. Non-backtracking exploration
  8. Random graphs.
  9. Rooted random trees.
  10. Backtracking exploration
  11. Random graphs.
  12. Rooted random trees.
  13. The multi-level friendship bias
  14. Short-level friendship bias for large sparse graphs
  15. Long-level friendship bias for large sparse graphs
  16. ...and 13 more sections

Key Result

Theorem 2.3

For each $k\in\mathbb N$, $\Delta_{[n]}^{(k)} \geq 0$, where for: The former holds for all $k$ if each connected component of $G_n$ is regular. The latter holds for odd $k$ if and only if each connected component of $G_n$ is regular, and for even $k$ if and only if each connected component of $G_n$ is either regular or bi-regular bipartite.

Figures (1)

  • Figure 1: Two examples of non-regular connected components for which equality in the $k$-level friendship paradox holds for $k=3$, respectively, $k=4$.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 4.1
  • ...and 19 more