81 problems

Persistent hub conjecture. If

Consider growing random graphs with infinite-variance degree distributions, often generated through a preferential-attachment mechanism. In the percolation critical window, the lar…

Rank-one random graph models with infinite-variance degree distributions are considered, under the assumption that they have no multiple edges. In the previously described critical…

Consider the undirected preferential attachment model with colors, and seek its local limit as a rooted random tree. A local-limit conjecture is that a potential candidate is a bra…

Consider the two-colored PAM with $m_1=m_2=m$, attachment kernel … and minority-vertex probability $\mu_1$. PageRank fairness conjecture. For every sufficiently small $c>0$, there…

Layer-size conjecture. For every $\alpha>-1$,

Cycle-limit conjecture. Distributional limits for the number of cycles of fixed length $2k$ or $2k+1$ should be Gaussian when $\delta>0$; when $\delta<0$, the limit should be

Fix $m\geqslant2$ and $\delta>0$. Let $T_n$ be the number of triangles in the instantaneous variant of the linear preferential attachment model, with connections specified by the i…

Oldest-vertex triangle decomposition conjecture. As $n\to\infty$, jointly,

Let $T_n(f)$ be the preferential attachment tree on $[n]$ generated by the nondecreasing attachment function $f:{\mathbb{N}}_0\to\mathbb{R}_{>0}$. Write $D_n(f)$ for the graph dist…

Prokhorenkova et al.'s modularity conjecture. With high probability,

Let the quantum preferential attachment (QPA) model have parameter $alpha>1$, and let the classical random (CR) model use parameter $r=1$ at the corresponding value of $alpha$. As…

Let $X$ denote the limiting random variable associated with the normalized maximum degree in Theorem 1.2, and let $D_1$ and $D_2$ be the constants appearing there. Maximum-degree l…

Let leaf-$\mathrm{PA}(a)$ be the growing random-tree model in which new vertices attach to existing vertices at rate $\ell+a$, where $\ell$ is the leaf degree and $a>0$. Let…

Let the simple graph $ath]\alpha$-dynamics be the process obtained from the $ath]\alpha$-dynamics by suppressing multiple edges, and let $t$ denote the number of included edges. Pi…

Let $m\ge 2$ and $\delta>0$, and let $G_n\sim \operatorname{PA}_n^{(m,\delta)}$. Let … Here $\nu$ is the exponential growth rate of the local weak limit of the preferential attachm…

Let the PAVD model be the preferential attachment model with vertex death from Definition. Suppose that the attachment and death-rate sequences $b$ and $d$ satisfy Assumptions and.…

Let $n$ denote the graph size and let $\beta$ be a model parameter. Suppose that the largest degree in a preferential attachment graph is of order $n^{\gamma(\beta)+o(1)}$. Let…

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…

Let $G_n$ be the final snapshot of a preferential attachment network on $n$ vertices, and suppose the changepoint is in the late-change regime $\tau_n=n-\Delta$ with…

Let $G_n^h=(V,E)$ be a preferential attachment graph, where $n$ is the number of vertices and $h$ is the number of edges added per step. Write $\operatorname{mod}(G_n^h)$ for its m…

Let $G_n^h=(V,E)$ be a preferential attachment graph, where $n$ is the number of vertices and $h$ is the number of edges added per step. Write $\operatorname{mod}(G_n^h)$ for its m…

Let $(\mathcal{T}_i)_{i\in\mathbb{N}_0}$ be a generalised preferential attachment tree, with random attachment-weight function values $(F(j))_{j\in\mathbb{N}_0}$. A unique persiste…

In a generalised preferential attachment model, new vertices attach to existing vertices with probability proportional to a positive function $f$ of the existing vertex's degree. A…

Let $G_n$ be a preferential attachment random graph of size $n$, and suppose its affine attachment parameter changes from $\delta_0$ to $\delta_1$ at the late change-point … where…