Cascading failures in networks of heterogeneous node behavior.

Variability in the dynamical function of nodes comprising a complex network impacts upon cascading failures that can compromise the network's ability to operate. Node types correspond to sources, sinks, or passive conduits of a current flow, applicable to renewable electrical power microgrids containing a variable number of intermittently operating generators and consumers of power. The resilience to cascading failures of ensembles of synthetic networks with different topology is examined as a function of the edge current carrying capacity and mix of node types, together with exemplar real-world networks. While a network with a homogeneous composition of node types can be resilient to failure, onewith an identical topology but with heterogeneous nodes can be strongly susceptible to failure. For networks with similar numbers of sources, sinks, and passive nodes the mean resilience decreases as networks become more disordered. Nevertheless all network topologies have enhanced regions of resilience, accessible by the manipulation of node composition and functionality.

Lacunarity of the zero crossings of Gaussian processes.

A lacunarity analysis of the zero crossings derived from Gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process. These bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a Poisson random process both possessing the same mean interval size between crossings. At very small and large scales compared with the correlation length of the random processes, the lacunarity is similar to the Poisson but exhibits significant departures from Poisson behavior if there is a zero-frequency component to the process's power spectrum. A comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias.

Price of anarchy on heterogeneous traffic-flow networks.

The efficiency of routing traffic through a network, comprising nodes connected by links whose cost of traversal is either fixed or varies in proportion to volume of usage, can be measured by the "price of anarchy." This is the ratio of the cost incurred by agents who act to minimize their individual expenditure to the optimal cost borne by the entire system. As the total traffic load and the network variability-parameterized by the proportion of variable-cost links in the network-changes, the behaviors that the system presents can be understood with the introduction of a network of simpler structure. This is constructed from classes of nonoverlapping paths connecting source to destination nodes that are characterized by the number of variable-cost edges they contain. It is shown that localized peaks in the price of anarchy occur at critical traffic volumes at which it becomes beneficial to exploit ostensibly more expensive paths as the network becomes more congested. Simulation results verifying these findings are presented for the variation of the price of anarchy with the network's size, aspect ratio, variability, and traffic load.

Enhanced twinkling in left-handed media.

A slab of left-handed material (LHM) with refractive index -1 forms a perfect lens that retains subwavelength information about a source or object. Such lenses are highly susceptible to perturbations affecting their performance. It is shown that illuminating a roughened interface between air and an LHM produces a regime for enhanced focusing of light close to the boundary. This generates caustics that are brighter, fluctuate more, and cause Gaussian speckle at distances closer to the interface than in right-handed matter. These effects present fresh challenges for perfecting the perfect lens.

Perfect lens with not so perfect boundaries.

In manufacturing left-handed media the interfaces will never be perfect; defects and other disturbances to interfaces and material parameters are unavoidable. We report an analytical calculation of electromagnetic wave propagation through a perfect lens with diffuse boundaries. Field localizations are generated in the boundary layers, and the lens' ability to recover evanescent modes in the presence of these boundaries is analyzed and quantified. It is shown that such a diffuse layer produces an effect that is qualitatively similar to a lens with increased losses.

Fluctuations in the zeros of differentiable Gaussian processes.

The stochastic point processes formed by the zero crossings or extremal points of differentiable, stationary Gaussian processes are studied as a function of their autocorrelation function. The properties of these point processes are mapped to the space formed by the parameters appearing in the autocorrelation function, their adopted form being sensitive to the structure of the autocorrelation function principally in the vicinity of the origin. The distribution for the number of zeros occurring in an asymptotically large interval are approximately negative-binomial or binomial depending upon whether the relative variance or Fano factor is greater or less than unity. The correlation properties of the zeros are such that they are repelled from each other or are "antibunched" if the autocorrelation function of the Gaussian process is characterized by a single scale size, but occur in clusters if more than one characteristic scale size is present. The intervals between zeros can be interpreted in terms of the autocorrelation function of the zeros themselves. When bunching occurs the interval density becomes bimodal, indicating the interval sizes within and between the clusters. The interevent periods are statistically dependent on one another with densities whose asymptotic behavior is governed by that of the autocorrelation function of the Gaussian process at large delay times. Poisson distributed fluctuations of the zeros occur only exceptionally but never form a Poisson process.

Continuous and discrete stable processes.

The one-sided Lévy-stable probability densities and the discrete-stable distributions form a doubly stochastic Poisson transform pair. This relationship facilitates the formulation of a class of continuous-stable stochastic processes.

Power-law-distributed level crossings define fractal behavior.

A relationship is established between the autocorrelation function of continuous Gaussian and non-Gaussian stochastic processes and the discrete process that describes their zero or level crossings. Random fractals occur when the distribution for the number of crossings is described by a class of Markov processes whose singlefold statistics are the discrete analog of the Lévy-stable continuous probability densities.

Fractional non-Brownian motion and trapping-time distributions of grains in rice piles.

Non-Gaussian height fluctuations occurring on the fueling time scale of a slowly driven rice pile match those observed in some turbulent/critical phenomena, forming an anticorrelated random fractal process with Hurst exponent H=0.2. Inspired by this observation, the concept of fractional Brownian motion (FBM) is extended to treat stochastic processes with skewed increments. Simulations of this process for antipersistent motion have first return time distribution deviating from the t(-2+H) law for FBM. The first return time distribution of this fractional non-Brownian motion describes and quantitatively determines the trapping-time distribution of grains in rice piles upon incorporating a continuous representation of the additional height fluctuations that occur on the time scale between fueling events.

Characterization of structural reorganization in rice piles.

Diagnostics applied to a rice-pile cellular automaton reveal different mechanisms producing power-law behaviors of statistical attributes of grains which are germane to self organised critical phenomena. The probability distributions for these quantities can be derived from two distinct random walk models that account for correlated clustered behavior through incorporating fluctuations in the number of steps in the walk. The first model describes the distribution for a spatial quantity, the resultant flight length of grains. This has a power-law tail caused by grains moving through a discrete, power-law distributed number of random steps of finite length. Developing this model into a random walk obtains distributions for the resultant flight length with characteristics similar to Lévy distributions. The second random walk model is devised to explain a temporal quantity, the distribution of "trapping" or "residence" times of grains at single locations in the pile. Diagnostics reveal that the trapping time can be constructed as a sum of "subtrapping times," which are described by a Lévy distribution where the number of terms in the sum is a discrete random variable accurately described by a negative binomial distribution. The infinitely divisible, two-parameter, limit distribution for the resultant of such a random walk is discussed, and describes a dual-scale power-law behavior if the number fluctuations are strongly clustered. The form for the distribution of transit times of grains results as a corollary.

Visibility depth improvement in active polarization imaging in scattering media.

A simple image-subtraction technique for further enhancement of the visibility depth in polarized imaging of surfaces immersed in scattering media is proposed and assessed. The technique is based on active illumination with circular or linear polarization states and image detection in the original and the opposite, or orthogonal, states. Contrast enhancement is achieved by subtraction of a fraction of the image recorded in the original state from that recorded in the opposite state. Results demonstrating the effectiveness of this method, obtained with Monte Carlo techniques, show that the visibility depth can be increased by as much as a mean free path. The results obtained are compared with those obtained by use of two alternative methods.

Lévy random walks with fluctuating step number and multiscale behavior.

Random walks with step number fluctuations are examined in n dimensions for when step lengths comprising the walk are governed by stable distributions, or by random variables having power-law tails. When the number of steps taken in the walk is large and uncorrelated, the conditions of the Lévy-Gnedenko generalization of the central limit theorem obtain. When the number of steps is correlated, infinitely divisible limiting distributions result that can have Lévy-like behavior in their tails but can exhibit a different power law at small scales. For the special case of individual steps in the walk being Gaussian distributed, the infinitely divisible class of K distributions result. The convergence to limiting distributions is investigated and shown to be ultraslow. Random walks formed from a finite number of steps modify the behavior and naturally produce an inner scale. The single class of distributions derived have as special cases, K distributions, stable distributions, distributions with power-law tails, and those characteristic of high and low frequency cascades. The results are compared with cellular automata simulations that are claimed to be paradigmatic of self-organized critical systems.