Edgar Gilbert

Edgar Nelson Gilbert (July 25, 1923 – June 15, 2013) was an American mathematician and

He died following a fall in 2013 at Basking Ridge, New Jersey.^[3]

Research

Coding theory

The

Central to the theory of random graphs is the Erdős–Rényi model, in which edges are chosen randomly for a fixed set of n vertices. It was introduced in two forms in 1959 by Gilbert, Paul Erdős, and Alfréd Rényi.^[G59][9] In Gilbert's G(n, p) form, each potential edge is chosen to be included in the graph or excluded from it, independently of the other edges, with probability p. Thus, the expected number of edges is pn(n − 1)/2, but the actual number of edges can vary randomly and all graphs have a nonzero probability of being selected. In contrast, in the G(n, M) model introduced by Erdős and Rényi, the graph is chosen uniformly at random among all M-edge graphs; the number of edges is fixed, but the edges are not independent of each other, because the presence of an edge in one position is negatively correlated with the presence of an edge in a different position. Although these two models end up having similar properties, the G(n, p) model is often more convenient to work with due to the independence of its edges.^[10]

In the mathematics of shuffling playing cards, the Gilbert–Shannon–Reeds model, developed in 1955 by Gilbert and Claude Shannon^[G55] and independently in unpublished work in 1981 by Jim Reeds, is a probability distribution on permutations of a set of n items that, according to experiments by Persi Diaconis, accurately models human-generated riffle shuffles. In this model, a deck of cards is split at a point chosen randomly according to a binomial distribution, and the two parts are merged with the order of merging chosen uniformly at random among all possible mergers. Equivalently, it is the inverse of a permutation formed by choosing independently at random for each card whether to put it into one of two piles (maintaining the original order of the cards within each pile), and then stacking the two piles on top of each other.^[11]

In 1961, Gilbert introduced the random plane network[13] (more commonly referred to now as a random geometric graph (RGG), or Gilbert Disk model) where points are placed on the infinite plane using a suitable Point Process and nodes connect if and only if they are within some critical connection range R; wireless communication networks were suggested as the main the application for this work. From this formulation a simple result follows that for a stationary Poisson point process in ${\displaystyle \mathbb {R} ^{2}}$ with density λ the expected degree of each node is the number of points found within the connectivity range, namely, πλR². A natural question to ask after formulating such a graph is what is the critical mean degree to ensure there is a giant component; in essence this question gave rise to the field of continuum percolation theory. By using a branching process Gilbert was able to provide an initial lower bound for the critical mean degree (equivalently the critical transmission range). By choosing an arbitrary point in the process (call this the zeroth generation), find all points within a connection distance R (first generation). Repeat the process for all points in the first generation ignoring any previously found and continue this process until it dies out. The associated branching process is one where the mean number of offspring is a Poisson random variable with intensity equal to the mean degree in the original RGG (πλR²). From here only standard branching process techniques need be applied to obtain a lower bound. Furthermore, Gilbert showed that by reframing the problem into one about bond percolation, an upper bound for the giant component can obtained. The method consists of discritizing the plane such that any two nodes in adjacent squares are connected; and allowing each square to represents an edge on the lattice. By construction, if there is a giant component in the bond percolation problem then there must be a giant component in the RGG.

Other contributions

Gilbert