We disprove a conjecture of Nagy on the maximum number of copies $N$($G$, $H$) of a fixed graph $G$ in a large graph $H$ with prescribed edge density. Nagy conjectured that for all $G$, the quantity $N$($G$, $H$) is asymptotically maximized by either a quasi-star or a quasi-clique. We show this is false for infinitely many graphs, the smallest of which has six vertices and six edges. We also propose some new conjectures for the behavior of $N$($G$, $H$) and present some evidence for them.

Secrecy graphs model the connectivity of wireless networks under secrecy constraints. Directed edges in the graph are present whenever a node can talk to another node securely in the presence of eavesdroppers. In the case of infinite networks, a critical parameter is the maximum density of eavesdroppers that can be accommodated while still guaranteeing an infinite component in the network, i.e., the percolation threshold. We focus on the case where the location of the nodes and the eavesdroppers are given by Poisson point processes, with and without power constraints. We present bounds for different types of percolation, including in-, out - and undirected percolation.

Deriving the critical density (which is equivalent to deriving the critical radius or power) to achieve coverage and/or connectivity for random deployments is a fundamental problem in the area of wireless networks. The probabilistic conditions normally derived, however, have limited appeal among practitioners because they areoften asymptotic, i.e., they only make high probability guarantees in the limit of large system sizes. Such conditions are not very useful in practice since deployment regions are always finite. Another major limitation of most existing work on coverage and connectivity is their focus on thick deployment regions (such as a square or a disk). There is no existing work (including traditional percolation theory) that derives critical densities for thin strips (or annuli).