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 \(\int _{\mathbb {R}}x\mu _n(\textrm{d}x) \ge 0\) , 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 Erdős-Rényi random graph, the inhomogeneous Erdős-Rényi 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 \(\mu ([0,\infty ))\ge \tfrac{1}{2}\) , a property we refer to as friendship paradox significance.
