12 problems

Let $N_R$ denote the number of refractory (R) elements at the end of the information-propagation process on the hierarchical network, and let $f(N_R)$ be its normalized frequency d…

Let $N$ be the system size, $p$ the fraction of rewired edges, and $ll(N,p)$ the characteristic path length. Let $N^*$ denote the $p$-dependent crossover length separating the regi…

Let $d_D$ denote the Deng information dimension and let $d_{dD}$ denote the d-summable Deng information dimension of a network. The d-summable Deng information dimension conjecture…

Let $n$ denote the network size, and consider the hidden-variable random graph model with its closure coefficient rescaled according to the asymptotic scaling studied in the paper.…

Let $c(\alpha,\nu)$ be the constant such that the fraction of vertices in the largest component of the KPKVB random graph $G(N;\alpha,\nu)$ converges in probability to…

The paper considers an exact susceptible-infected-susceptible epidemic model and a sufficient condition given earlier in the paper, denoted by … may be relaxed. The conjecture conc…

Circular law conjecture. Under suitable boundedness assumptions on the moments of $z$, $\mu_n$ converges to $\mu_\infty$.

Triangle phase-transition conjecture. The sharp phase transition in the probability of triangles occurs at

Geodesic-distance conjecture. The difference between the $1{,}000$-node and $10{,}000$-node lattice networks is caused by the average geodesic distance: the average node distance f…

Let a small-world network be compared with a regular lattice, and consider the network dynamics on these networks, with algebraic connectivity serving as a measure of convergence o…

A competitive percolation process is a process in which links are added under competition between clusters; let $p_{\mathsf{gr}}$ denote the probability that the largest cluster gr…

Let a complex tree be a scale-invariant network with tree structure, and let $D_b$ denote its box dimension and $D_w$ its walk exponent. The mean round trip time averaged over pair…