21 problems

Inverse-square variance conjecture.

Bounded-variance conjecture. As $v$ tends to infinity,

Asymptotic shape conjecture. There exists a nonempty compact subset $B_0$ of $\mathbb{R}^2$ such that, for every positive number $\varepsilon$,

Let a nonequilibrium steady state (NESS) on a dense network have entropy production rate $\sigma_{\mathrm{NESS}}$, and let … be the minimal entropy production over probability dist…

Finiteness conjecture. In certain planar network models, all classical trajectories are almost surely finite for every positive disorder parameter.

The models under consideration are random network ensembles whose bulk spectral density has a singularity, such as the Poisson random graph with Gaussian couplings or the sparse Ba…

Recursive limit-law conjecture. The limit law of $F$ is determined recursively as follows. If $P$ is a normal pattern, then $F$ is a Poisson pattern; in all other cases for $P$,…

Let a multipartite graphon be a graphon whose degree function $\operatorname{Deg}(x)$ is piecewise constant. The preceding random-network results concern steady-state pattern-formi…

Let $\mathbb{T}^d$ be the $d$-regular tree, where $d\geq 3$, and let $p_c=1/(d-1)$ and $\lambda_c=d-1$. Critical recurrence conjecture. For any $d\geq 3$, the biased network…

Let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ be a biased disordered random network, with $d\geq 3$ and $0<\lambda_1\leq 1<\lambda_2$. A network is…

Let $p_c$ denote the critical probability for Bernoulli bond percolation on ${\mathbb{Z}}^d$, and let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ den…

Let a high-dimensional random Apollonian network (HDRAN) evolve with network index $k$, and let its Wiener index be the sum of the distances between all unordered pairs of vertices…

Consider a network built dynamically from randomly arriving Poisson points, with edges created as line segments attaching each arriving point to the existing network by a scale-inv…

Multilayer random-network conjecture. Under these conditions,

Random-network ideality conjecture. As $L_1$ and $L_2$ increase, for $L_1/L_2>1$ and arbitrary $p$, the probability that the final estimate is ideal satisfies

High-variance network conjecture. For the final estimate to have large variance, some upstream agents should have a disproportionately large out-degree, while the remaining agents…

Let $\Pi$ be the planar Poisson line process, and let the network generated by $\Pi$ consist of the geodesic routes between points in an independent Poisson point process $\Xi$ in…

Final-size conjecture. Subject to mild conditions on the household size and global degree distributions, the expected relative final size of a major outbreak converges to the survi…

Maintenance-cost robustness conjecture. Allowing $v_1$ to be large and negative to account for agents foreseeing the maintenance cost would not affect the analysis of the high-inte…

High-intensity unilateral stability conjecture. In the high-intensity regime, the network is asymptotically almost surely unilaterally stable for any finite $\tilde{c}$.

Sufficiency conjecture. If, for all bounded $A,B$,