Moment criterion conjecture for ultra-small distances

Let $\mathbb{G}_n$ and the uniformly chosen vertices $U_{n,1},U_{n,2}$ be as in Theorem

, and let $D$ denote the limiting degree distribution. **Moment criterion conjecture.** Under the assumptions of Theorem

, there is a constant $\varepsilon_{\ast}\in(0,1)$ depending on the model parameters such that, for any $\varepsilon>\varepsilon_{\ast}$, if $\mathbb{E}[D^{2-\varepsilon}]=\infty$, then there is a constant $C(\alpha,d)>0$ such that

$$\mathbb{P}\left(d_{\mathbb{G}_n}(U_{n,1},U_{n,2})<C(\alpha,d)\log\log n\mid U_{n,1}\text{ and }U_{n,2}\text{ are connected in }\mathbb{G}_n\right)\to1,$$

as $n\to\infty$. This conjecture proposes that divergence of a sufficiently low-order moment universally implies ultra-small, doubly logarithmic distances; the source notes that this behavior is observed for a special class of models, whereas larger-than-doubly-logarithmic distances can also occur when the second moment is infinite.