Extreme robustness of scaling in sample space reducing processes explains Zipf's law in diffusion on directed networks

It has been shown recently that a specific class of path-dependent stochastic processes, which reduce their sample space as they unfold, lead to exact scaling laws in frequency and rank distributions. Such Sample Space Reducing processes (SSRP) offer an alternative new mechanism to understand the emergence of scaling in countless processes. The corresponding power law exponents were shown to be related to noise levels in the process. Here we show that the emergence of scaling is not limited to the simplest SSRPs, but holds for a huge domain of stochastic processes that are characterized by non-uniform prior distributions. We demonstrate mathematically that in the absence of noise the scaling exponents converge to $-1$ (Zipf's law) for almost all prior distributions. As a consequence it becomes possible to fully understand targeted diffusion on weighted directed networks and its associated scaling laws law in node visit distributions. The presence of cycles can be properly interpreted as playing the same role as noise in SSRPs and, accordingly, determine the scaling exponents. The result that Zipf's law emerges as a generic feature of diffusion on networks, regardless of its details, and that the exponent of visiting times is related to the amount of cycles in a network could be relevant for a series of applications in traffic-, transport- and supply chain management.Comment: 11 pages, 5 figure

Data-Driven Time-Frequency Analysis

In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and non-stationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form $\{a(t) \cos(\theta(t))\}$, where $a \in V(\theta)$, $V(\theta)$ consists of the functions smoother than $\cos(\theta(t))$ and $\theta'\ge 0$. This problem can be formulated as a nonlinear $L^0$ optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the $L^0$ optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide a convergence analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the EMD/EEMD method. We also apply our method to study data without scale separation, data with intra-wave frequency modulation, and data with incomplete or under-sampled data

Objective and efficient terahertz signal denoising by transfer function reconstruction

As an essential processing step in many disciplines, signal denoising efficiently improves data quality without extra cost. However, it is relatively under-utilized for terahertz spectroscopy. The major technique reported uses wavelet denoising in the time-domain, which has a fuzzy physical meaning and limited performance in low-frequency and water-vapor regions. Here, we work from a new perspective by reconstructing the transfer function to remove noise-induced oscillations. The method is fully objective without a need for defining a threshold. Both reflection imaging and transmission imaging were conducted. The experimental results show that both low- and high-frequency noise and the water-vapor influence were efficiently removed. The spectrum accuracy was also improved, and the image contrast was significantly enhanced. The signal-to-noise ratio of the leaf image was increased up to 10 dB, with the 6 dB bandwidth being extended by over 0.5 THz

THz generation using a reflective stair-step echelon

We present a novel method for THz generation in lithium niobate using a reflective stair-step echelon structure. The echelon produces a discretely tilted pulse front with less angular dispersion compared to a high groove-density grating. The THz output was characterized using both a 1-lens and 3-lens imaging system to set the tilt angle at room and cryogenic temperatures. Using broadband 800 nm pulses with a pulse energy of 0.95 mJ and a pulse duration of 70 fs (24 nm FWHM bandwidth, 39 fs transform limited width), we produced THz pulses with field strengths as high as 500 kV/cm and pulse energies as high as 3.1 $\mu$J. The highest conversion efficiency we obtained was 0.33%. In addition, we find that the echelon is easily implemented into an experimental setup for quick alignment and optimization.Comment: 19 pages, 4 figure