Georgie Knight

Follow the fugitive: an application of the method of images to open systems

Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open systems. As an example, we explicitly derive the exact size- and position-dependent escape rate in a Markov case for holes of finite-size. Moreover, a general relation between the transfer operators of the closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in many areas of the physical sciences such as information theory, network theory, quantum Weyl law and, via Ulam?s method, can be used as an approximation method in general dynamical systems.

Capturing correlations in chaotic diffusion by approximation methods.

We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line that contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher-order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, while the third method approximates Markov partitions and transition matrices by using a slight variation of the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence, and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in the case of dynamics where exact results for the diffusion coefficient are not available.

Counting Geodesic Paths in 1-D VANETs

In the IEEE 802.11p standard addressing vehicular communications, basic safety messages can be bundled together and relayed as to increase the effective communication range of transmitting vehicles. This process forms a vehicular ad hoc network (VANET) for the dissemination of safety information. The number of “shortest multihop paths” (or geodesics) connecting two network nodes is an important statistic which can be used to enhance throughput, validate threat events, protect against collusion attacks, infer location information, and also limit redundant broadcasts thus reducing interference. To this end, we analytically calculate for the first time the mean and variance of the number of geodesics in 1-D VANETs.

Temporal-varying failures of nodes in networks.

We consider networks in which random walkers are removed because of the failure of specific nodes. We interpret the rate of loss as a measure of the importance of nodes, a notion we denote as failure centrality. We show that the degree of the node is not sufficient to determine this measure and that, in a first approximation, the shortest loops through the node have to be taken into account. We propose approximations of the failure centrality which are valid for temporal-varying failures, and we dwell on the possibility of externally changing the relative importance of nodes in a given network by exploiting the interference between the loops of a node and the cycles of the temporal pattern of failures. In the limit of long failure cycles we show analytically that the escape in a node is larger than the one estimated from a stochastic failure with the same failure probability. We test our general formalism in two real-world networks (air-transportation and e-mail users) and show how communities lead to deviations from predictions for failures in hubs.

Linear and fractal diffusion coefficients in a family of one-dimensional chaotic maps

We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation. Using a technique by which Taylor–Green–Kubo formulae are evaluated in terms of generalized Takagi functions, we derive exact, fully analytical expressions for the diffusion coefficients. Typically, for simple maps these quantities are fractal functions of control parameters. However, our family of four maps exhibits both fractal and linear behaviour. We explain these different structures by looking at the topology of the Markov partitions and the ergodic properties of the maps.

Symmetric motifs in random geometric graphs

We study symmetric motifs in random geometric graphs. Symmetric motifs are subsets of nodes which have the same adjacencies. These subgraphs are particularly prevalent in random geometric graphs and appear in the Laplacian and adjacency spectrum as sharp, distinct peaks, a feature often found in real-world networks. We look at the probabilities of their appearance and compare these across parameter space and dimension. We then use the Chen-Stein method to derive the minimum separation distance in random geometric graphs which we apply to study symmetric motifs in both the intensive and thermodynamic limits. In the thermodynamic limit the probability that the closest nodes are symmetric approaches one, whilst in the intensive limit this probability depends upon the dimension.

Sparre-Andersen theorem with spatiotemporal correlations.

The Sparre-Andersen theorem is a remarkable result in one-dimensional random walk theory concerning the universality of the ubiquitous first-passage-time distribution. It states that the probability distribution ρ(n) of the number of steps needed for a walker starting at the origin to land on the positive semiaxes does not depend on the details of the distribution for the jumps of the walker, provided this distribution is symmetric and continuous, where in particular ρ(n) ∼ n(-3/2) for large number of steps n. On the other hand, there are many physical situations in which the time spent by the walker in doing one step depends on the length of the step and the interest concentrates on the time needed for a return, not on the number of steps. Here we modify the Sparre-Andersen proof to deal with such cases, in rather general situations in which the time variable correlates with the step variable. As an example we present a natural process in 2D that shows that deviations from normal scaling are present for the first-passage-time distribution on a semiplane.