6 problems

Let the finite $d$-dimensional grid be the wrapped-around grid graph with $n$ nodes. For a set $S$ of $j$ nodes, let $E(S)$ denote the set of incoming edges of $S$, and suppose…

Let $\mathbb{K \cap G}(n;\pmb{\Theta}_n)$ denote the intersection of an inhomogeneous random key graph and an Erdős–Rényi graph, with class probability distribution…

Penrose's isolated-node conjecture. As $\rho\to\infty$, the total number of isolated nodes is well approximated by a Poisson distribution with mean

Let $\Delta(k)$ denote the difference between the probability that a network has minimum degree at least $k$ and the probability that it is fully connected. For $k\geq 1$, consider…

Isotropy-removal conjecture. Under some mild conditions, for example when nodes are independently and randomly oriented, assumption a) can be removed while the paper's results on a…

Let $n$ sensor nodes be distributed in a disk of unit area, with transmission range $\rho_n$, and let $\beta_n$ denote the probability that two nodes in the Eschenauer–Gligor key d…