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

Secrecy Coverage (Conference Proceeding)

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→ ∞

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→∞

The Linus Sequence

Define the Linus sequence Ln for n ≥ 1 as a 0–1 sequence with L1 = 0, and Ln chosen so as to minimize the length of the longest immediately repeated block Ln−2r+1 Ln−r = Ln−r+1 Ln. Define the Sally sequence Sn as the length r of the longest repeated block that was avoided by the choice of Ln. We prove several results about these sequences, such as exponential decay of the frequency of highly periodic subwords of the Linus sequence, zero entropy of any stationary process obtained as a limit of word frequencies in the Linus sequence and infinite average value of the Sally sequence. In addition we make a number of conjectures about both sequences

Paths in Graphs

We prove that if 10 ≦ (k2) ≦ m \u3c (k+12) then the number of paths of length three in a graph G of size m is at most 2m(m – k)(k - 2)/k. Equality is attained if G is the union of Kk and isolated vertices. We also give asymptotically best possible bounds for the maximal number of paths of length s, for arbitrary s, in graphs of size m. Lastly, we discuss the more general problem of maximizing the number of subgraphs isomorphic to a given graph H in graphs of size m

Connectivity of Random k-Nearest-Neighbor Graphs

Let P be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of Pto its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074logn then the probability that Gn,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774logn, then the probability that Gn,k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1))logn. In this paper we improve these lower and upper bounds to 0.3043logn and 0.5139logn, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209logn and 0.9967logn for the directed version of this problem. A related question concerns coverage. With Gn,k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209logn then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967logn, then the probability that the discs cover Sn tends to 1 as n → ∞

Sentry Selection in Wireless Networks

Let P be a Poisson process of intensity one in the infinite plane R2. We surround each point x of P by the open disc of radius r centred at x. Now let Sn be a fixed disc of area n, and let Cr(Sn) be the set of discs which intersect Sn. Write Erk for the event that Cr(Sn) is a k-cover of Sn, and Frk for the event that Cr(Sn) may be partitioned into k disjoint single covers of Sn. We prove that P(Erk ∖ Frk) ≤ ck / logn, and that this result is best possible. We also give improved estimates for P(Erk). Finally, we study the obstructions to k-partitionability in more detail. As part of this study, we prove a classification theorem for (deterministic) covers of R2 with half-planes that cannot be partitioned into two single covers