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Let $G_n$ be an undirected finite graph on $n\in\mathbb{N}$ vertices labelledby $[n] = \{1,\ldots,n\}$. For $i \in [n]$, let $\Delta_{i,n}$ be thefriendship bias of vertex $i$, defined as the difference between the averagedegree 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 thefriendship-bias empirical distribution, i.e., the measure that puts mass$\frac{1}{n}$ at each $\Delta_{i,n}$, $i \in [n]$. The friendship paradox saysthat if $G_n$ has no self-loops, then $\int_{\mathbb{R}} x\mu_n(\mathrm{d}x)\geq 0$, with equality if and only if in each connected component of $G_n$ allthe degrees are the same.  We show that if $(G_n)_{n\in\mathbb{N}}$ is a sequence of sparse randomgraphs that converges to a rooted random tree in the sense of convergencelocally 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. Westudy $\mu$ for four classes of sparse random graphs: the homogeneousErd\H{o}s-R\'enyi random graph, the inhomogeneous Erd\H{o}s-R\'enyi randomgraph, the configuration model and the preferential attachment model. Inparticular, we compute the first two moments of $\mu$, identify the right tailof $\mu$, and argue that $\mu([0,\infty))\geq\tfrac{1}{2}$, a property we referto as friendship-paradox significance.

The friendship paradox for sparse random graphs