Percolation in the Secrecy Graph

The secrecy graph is a random geometric graph which is intended to 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, which, in the model, is determined solely by the locations of the nodes and 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 locations of the nodes and eavesdroppers are given by Poisson point processes, and present bounds for different types of percolation, including in-, out- and undirected percolation.Comment: 22 pages, 3 figure

Continuum AB percolation and AB random geometric graphs

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda,\mu$. We show for $d \geq 2$ that if $\lambda$ is supercritical for the one-type random geometric graph with distance parameter $2r$, there exists $\mu$ such that $(\lambda,\mu)$ is supercritical (this was previously known for $d=2$). For $d=2$ we also consider the restriction of this graph to points in the unit square. Taking $\mu = \tau \lambda$ for fixed $\tau$, we give a strong law of large numbers as $\lambda \to \infty$, for the connectivity threshold of this graph

On the critical threshold for continuum AB percolation

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c/(λ-λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞

Secrecy Coverage

Motivated by information-theoretic secrecy, geometric models for secrecy in wireless networks have begun to receive increased attention. The general question is how the presence of eavesdroppers affects the properties and performance of the network. Previously, the focus has been mostly on connectivity. Here we study the impact of eavesdroppers on the coverage of a network of base stations. The problem we address is the following. Let base stations and eavesdroppers be distributed as stationary Poisson point processes in a disk of area n. If the coverage of each base station is limited by the distance to the nearest eavesdropper, what is the maximum density of eavesdroppers that can be accommodated while still achieving full coverage, asymptotically as n→∞

On the critical threshold for continuum AB percolation

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda,\mu$. For any $\lambda>0$ we consider the percolation threshold $\mu_c(\lambda)$ associated to the parameter $\mu$. Denoting by $\lambda_c:= \lambda_c(2r)$ the percolation threshold for the standard Poisson Boolean model with radii $r$, we show the lower bound $\mu_c(\lambda)\ge c\log(c/(\lambda-\lambda_c))$ for any $\lambda>\lambda_c$ with $c>0$ a fixed constant. In particular, $\mu_c(\lambda)$ tends to infinity when $\lambda$ tends to $\lambda_c$ from above.Comment: 12 pages, 3 figure

Paths and cycles in graphs and hypergraphs

In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles. A $k$-uniform tight cycle $C^{(k)}_n$ is a $k$-uniform hypergraph on $n$ vertices with a cyclic ordering of its vertices such that the edges are all $k$-sets of consecutive vertices in the ordering. We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to $k$-uniform hypergraphs and prove results in the 4- and 5-uniform case. For a $k$-uniform hypergraph~$H$, the Ramsey number ${r(H)}$ is the smallest integer $N$ such that any 2-edge-colouring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that $r(C^{(4)}_n)$ = (5+$o$(1))$n$. We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any $\gamma$ >0 and $k$ $\geq$ 3 asymptotically almost surely, every subgraph of the binomial random $k$-uniform hypergraph $G^{(k)}(n, n^{\gamma -1})$ in which all $(k-1)$-sets are contained in at least $(\frac{1}{2}+2\gamma)pn$ edges has a tight Hamilton cycle. A random graph model on a host graph $H$ is said to be 1-independent if for every pair of vertex-disjoint subsets $A,B$ of $E(H)$, the state of edges (absent or present) in $A$ is independent of the state of edges in $B$. We show that $p$ = 4 - 2$\sqrt{3}$ is the critical probability such that every 1-independent graph model on $\mathbb{Z}^2 \times K_n$ where each edge is present with probability at least $p$ contains an infinite path