10 problems

Let $G_n$ be the random graph and let $o_1,o_2$ be the two vertices whose distance is considered. Assume the hypotheses of Theorem. Then typical distance lower-bound conjecture. Th…

For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ be the smallest integer such that every graph $G$ with at least $h_t(\Delta)$ edges and maximum degree…

For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ be the smallest integer such that every graph $G$ with at least $h_t(\Delta)$ edges and maximum degree…

Let $m\geq 2$ and let $\delta>0$ in a preferential attachment model (PAM), and let $\nu\in(1,\infty)$ denote the corresponding effective branching factor. The diameter is the large…

Consider scale-free percolation in the polylogarithmic regime, with parameters $\tau>3$ and $\alpha\in(1,2)$, and let $\Delta(\alpha)$ denote the exponent governing graph distances…

Consider scale-free percolation (SFP), the geometric inhomogeneous random graph (GIRG), and the hyperbolic random graph (HRG), and let $\mathcal{B}(x,k)$ denote the graph-distance…

Let $x_n,y_n$ be two uniform, independent vertices of $\mathbf{T}_{2n,g_n}$. Let $m_{\theta}\in(0,1)$ be the parameter defined by the preceding relation between $h_{\theta}$ and…

Let $\mathbb{G}_n$ and the uniformly chosen vertices $U_{n,1},U_{n,2}$ be as in Theorem … , there is a constant $\varepsilon_{\ast}\in(0,1)$ depending on the model parameters such…

Incident-edge conjecture. If every side of every blob has enough incident edges, then the network should be uniquely reconstructible from its shortest distances.

Logarithmic-threshold conjecture. The distance formula in the paper should remain valid when