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 $Δ_{i,n}$ 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 $μ_n$ denote the friendship-bias empirical distribution, i.e., the measure that puts mass $\frac{1}{n}$ at each $Δ_{i,n}$, $i \in [n]$. The friendship paradox says that $\int_{\mathbb{R}} xμ_n(\mathrm{d}x) \geq 0$, 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 $μ_n$ converges weakly to a limiting measure $μ$ that is expressible in terms of the law of the rooted random tree. We study $μ$ 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 $μ$, identify the right tail of $μ$, and argue that $μ([0,\infty))\geq\tfrac{1}{2}$, a property we refer to as friendship paradox significance.

Figures (4)

• Figure 1: Numerical plot of the map $\lambda\mapsto \sum_{k=0}^{10^{4}} \tfrac{\mathrm{e}^{-\lambda}\lambda^{k}}{k!} \sum_{l\geq k(k-1)} \tfrac{\mathrm{e}^{-\lambda k}(\lambda k)^{l}}{l!}$ for $10^{-8} \leq \lambda \leq 10^3$ for $\mathrm{HER}_{n}(\frac{\lambda}{n} \wedge 1)$.
• Figure 2: Numerical plot of the map $\lambda\mapsto \mu([0,\infty))$ for $\mathrm{IER}_{n}(\lambda f)$ with an increasing, a decreasing and a non-monotonic function $f$, estimated with the help of Monte Carlo integration.
• Figure 3: Numerical plot of the region containing the map $\tau\mapsto \mu([0,\infty))$ for $\mathrm{CM}_{n}(\mathbf{d}_{n})$ with $p_{k} = k^{-\tau}/\zeta(\tau)$.
• Figure 4: Numerical plot of a lower bound on $\delta\mapsto \mu([0,\infty))$ for $(\mathrm{PAM}_{n}^{(1,\delta)})_{n\in\mathbb N}$.

Theorems & Definitions (22)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Definition 2.5
• Conjecture 2.6
• proof : Proof of Theorem \ref{['thm1']}
• proof : Proof of Theorem \ref{['thm2']}
• Theorem 4.1
• Theorem 4.2