The friendship paradox for sparse random graphs

Abstract

Let $G_n$ be an undirected finite graph on $n\in\mathbb{N}$ vertices labelled by $[n] = \{1,\ldots,n\}$. For $i \in [n]$, let $\Delta_{i,n}$ be the friendship bias of vertex $i$, defined as the difference between the average degree of the neighbours of vertex $i$ and the degree of vertex $i$ itself when $i$ is not isolated, and zero when $i$ is isolated. Let $\mu_n$ denote the friendship-bias empirical distribution, i.e., the measure that puts mass $\frac{1}{n}$ at each $\Delta_{i,n}$, $i \in [n]$. The friendship paradox says that $\int_{\mathbb{R}} x\mu_n(\mathrm{d}x) \geq 0$, with equality if and only if in each connected component of $G_n$ all the degrees are the same. We show that if $(G_n)_{n\in\mathbb{N}}$ is a sequence of sparse random graphs that converges to a rooted random tree in the sense of convergence locally in probability, then $\mu_n$ converges weakly to a limiting measure $\mu$ that is expressible in terms of the law of the rooted random tree. We study $\mu$ for four classes of sparse random graphs: the homogeneous Erd\H{o}s-R\'enyi random graph, the inhomogeneous Erd\H{o}s-R\'enyi random graph, the configuration model and the preferential attachment model. In particular, we compute the first two moments of $\mu$, identify the right tail of $\mu$, and argue that $\mu([0,\infty))\geq\tfrac{1}{2}$, a property we refer to as friendship paradox significance.