Random Walk Process Research Articles

Gravitational wave memory imprints on the CMB from populations of massive black hole mergers

Aims. Our aim is to showcase and characterise the rich phenomenology of temperature fluctuation patterns that are imprinted on the cosmic microwave background (CMB) by the gravitational wave memory (GWM) of massive black hole mergers. Methods. We analysed both individual binaries as well as populations of binaries, distributed in local cosmological boxes at a given redshift. Results. The magnitude of the temperature fluctuations scales primarily as a function of binary total mass and pattern angular scale, and accumulates as a random-walk process when populations of mergers are considered. Fluctuations of order ∼10−12 K are reached across scales of ∼1′ to ∼1° for realistic volumetric merger rates of 10−3 Mpc−3 Gyr−1, as is appropriate for massive galaxies at z = 1. We determined numerically that GWM temperature fluctuations result in a universal power spectrum with a scaling of P(k)∝k−2.7. Conclusions. While not detectable given the limitations of current all-sky CMB surveys, our work explicitly shows how every black hole merger in the Universe left us its unique faint signature.

Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Superdiffusion.

Superdiffusion is usually defined as a random walk process of a molecule, in which the time evolution of the mean-squared displacement, σ2, of the molecule is a power function of time, σ2(t)∼t2/γ, with γ∈(1,2). An equation with a Riesz-type fractional derivative of the order γ with respect to a spatial variable (a fractional superdiffusion equation) is often used to describe superdiffusion. However, this equation leads to the formula σ2(t)=κt2/γ with κ=∞, which, in practice, makes it impossible to define the parameter γ. Moreover, due to the nonlocal nature of this derivative, it is generally not possible to impose boundary conditions at a thin partially permeable membrane. We show a model of superdiffusion based on an equation in which there is a fractional Caputo time derivative with respect to another function, g; the spatial derivative is of the second order. By choosing the function in an appropriate way, we obtain the g-superdiffusion equation, in which Green's function (GF) in the long time limit approaches GF for the fractional superdiffusion equation. GF for the g-superdiffusion equation generates σ2 with finite κ. In addition, the boundary conditions at a thin membrane can be given in a similar way as for normal diffusion or subdiffusion. As an example, the filtration process generated by a partially permeable membrane in a superdiffusive medium is considered.

A novel high applicability link prediction based on biased random walks

Complex systems are prevalent in nature, and link prediction is crucial for analyzing these systems. This method has gained attention for its ability to uncover various irregular movements with underlying similarities. While individual particle behavior is unpredictable, the collective behavior of large groups can be forecasted more accurately. The variability at lower scales vs universal similarities at higher scales is often overlooked. To address this, we investigate the random walk process on directed networks at a global scale. We propose an improved link prediction method using biased random walks to enhance accuracy and applicability. We first define out-degree neighbors as valid transmission options to prevent conflicts with in-degree paths. We then analyze how out-degrees and in-degrees affect particle transition probabilities, guiding particles toward high out-degrees and low in-degrees for a sustainable process. Finally, we use particle stabilization probabilities at different nodes as a similarity measure for predicting potential connections. Our method improves prediction precision and practicality, as validated by experiments on nine real-world networks, showing significant gains in accuracy and AUC scores.

Unifying suspension and granular shear-induced self-diffusion

Shear-induced self-diffusion is a fundamental mode of transport in granular flows. Yet its critical behaviour and dependence on the particle solid fraction are still unclear. Here, we rationalize these dependencies by performing two-dimensional pressure-imposed numerical simulations of dense non-Brownian frictional suspensions. Our results, combined with existing numerical data on inertial granular flows, show that the shear-induced diffusion coefficients of both systems can be captured by a single function of the distance to jamming. They further show that the grain diffusive behaviour is underpinned by a specific random walk process, having a constant elementary step length driven at a frequency that increases with the solid fraction. The proposed scaling laws pave the way for a better understanding of mixing processes in granular media.

Quantum phenomena in biological systems

Quantum biology is a modern field of research that aims to understand how quantum effects can affect the chemistry underlying various biological processes. This paper reviews several examples of biological processes where quantum effects might play a notable role. Initially, the photon capture mechanism present in vision is discussed, where the energy of the photon is used to cause conformational changes to chromophoric proteins. The second example elaborates the highly efficient energy transfer process present in photosynthesis and discusses, in particular, how the random quantum walk process may enhance the performance drastically. Subsequently, the vertebrate magnetoreception, and the possible associated role of the radical pair mechanism in the process is considered. The review concludes with the discussion of some speculative ideas of putative quantum effects arising in neural processes.

Improved GPS tropospheric path delay estimation using variable random walk process noise

Accurate positioning using the Global Positioning System relies on accurate modeling of tropospheric delay. Estimated tropospheric delay must vary sufficiently to capture true variations; otherwise, systematic errors propagate into estimated positions, particularly the vertical. However, if the allowed delay variation is too large, the propagation of data noise into all parameters is amplified, reducing precision. Here we investigate the optimal choice of tropospheric constraints applied in the GipsyX software, which are specified by values of random walk process noise. We use the variability of 5-min estimated positions as a proxy for tropospheric error. Given that weighted mean 5-min positions closely replicate 24-h solutions, our ultimate goal is to improve 24-h positions and other daily products, such as precise orbit parameters. The commonly adopted default constraint for the zenith wet delay (ZWD) is 3 mm/√(hr) for 5-min data intervals. Using this constraint, we observe spurious wave-like patterns of 5-min vertical displacement estimates with amplitudes ~ 100 mm coincident with Winter Storm Ezekiel of November 27, 2019, across the central/eastern USA. Loosening the constraint suppresses the spurious waves and reduces 5-min vertical displacement variability while improving water vapor estimates. Further improvement can be achieved when optimizing constraints regionally, or for each station. Globally, results are typically optimized in the range of 6–12 mm/√(hr). Generally, we at least recommend loosening the constraint from the current default of 3 mm/√(hr) to 6 mm/√(hr) for ZWD every 300 s. Constraint values must be scaled by √(x/300) for alternative data intervals of x seconds.

Does speculation impair informational efficiency? New evidence from the Indian agricultural commodity market

Purpose This study explores the association between market efficiency and speculation. The government of India temporarily banned the futures trading of various commodities several times citing the presence of speculation. Many controversies exist about this topic; thus, this study clarifies the association between market efficiency and speculation and investigates whether market reforms altered this association.Design/methodology/approach The data for nine commodities is collected from the National Commodity and Derivative Exchange (NCDEX) for 2005–2022. Regression analysis and Automatic Variance Ratio (AVR) were adopted to inspect the informational efficiency and influence of speculation in the commodity market. Furthermore, this study uses different sub-samples to understand the changes in the market microstructure and its effects on market quality.Findings The results confirm an inverse and significant relationship between information efficiency and speculation and a deviation from the random walk process observed. Therefore, return predictability exists in the market. This study confirms that market reforms do not reduce the influence of speculation on market efficiency. The study concludes that the market is not weak-form efficient.Research limitations/implications This study has certain limitations, since this study is empirical in nature, it may possess the limitations of empirical research.Originality/value This paper has dual novelty. First, this study investigates the effects of market reforms. Second, this study captures the influence of speculation in the Indian agricultural commodity market by considering the market microstructure aspects.

Outlier detection using local density and global structure

Outlier detection using local density and global structure

Identification of Drought Stress-Responsive Genes in Rice by Random Walk with Multi-Restart Probability on MultiPlex Biological Networks.

Exploring drought stress-responsive genes in rice is essential for breeding drought-resistant varieties. Rice drought resistance is controlled by multiple genes, and mining drought stress-responsive genes solely based on single omics data lacks stability and accuracy. Multi-omics correlation analysis and biological molecular network analysis provide robust solutions. This study proposed a random walk with a multi-restart probability (RWMRP) algorithm, based on the Restarted Random Walk (RWR) algorithm, to operate on rice MultiPlex biological networks. It explores the interactions between biological molecules across various levels and ranks potential genes. RWMRP uses eigenvector centrality to evaluate node importance in the network and adjusts the restart probabilities accordingly, diverging from the uniform restart probability employed in RWR. In the random walk process, it can be better to consider the global relationships in the network. Firstly, we constructed a MultiPlex biological network by integrating the rice protein-protein interaction, gene pathway, and gene co-expression network. Then, we employed RWMRP to predict the potential genes associated with rice tolerance to drought stress. Enrichment and correlation analyses resulted in the identification of 12 drought-related genes. We further conducted quantitative real-time polymerase chain reaction (qRT-PCR) analysis on these 12 genes, ultimately identifying 10 genes responsive to drought stress.

On the Boundary Functional of a Semi-Markov Process

In this paper, we consider the semi-Markov random walk process with negative drift, positive jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary functional is obtained. In this work, we define the residence time of the system by generalized exponential distributions with different parameters via fractional order integral equation. The purpose of this paper is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary functional of the semi-Markov random walk processes to fractional order differential equation with constant coefficients.

Rate of Weak Convergence of Random Walk with a Generalized Reflecting Barrier

In this study, a random walk process with generalized reflecting barrier is considered and an inequality for rate of weak convergence of the stationary distribution of the process of interest is propounded. Under some conditions, the weak convergence theorem for the stationary distribution of the considered process is obtained in literature, however, the rate of the convergence is not studied sufficiently. Nonetheless, one of the most crucial issues in probability theory is the convergence rate in limit theorems, as it affects the precision and effectiveness of using these theorems in practice. Therefore, for the rate of convergence for the examined process, comparatively simple inequality is represented. The obtained inequality demonstrates that the rate of convergence is correlated with the tail of the distribution of ladder heights of the random walk.

A New Timescale–Mass Scaling for the Optical Variation of Active Galactic Nuclei across the Intermediate-mass to Supermassive Scales

The variability of active galactic nuclei (AGNs) has long been servicing as an essential avenue of exploring the accretion physics of a black hole (BH). There are two commonly used methods for analyzing AGN variability. First, the AGN variability, characterized by the structure function (SF) of a single band, can be well described by a damped random walk (DRW) process on timescales longer than ∼weeks, shorter than which departures have been reported. Second, the color variation (CV) between two bands behaves timescale dependent, raising challenges to the widely accepted reprocessing scenario. However, both the departure from the DRW process and the timescale-dependent CV are hitherto limited to AGNs, mainly quasars, at the supermassive scale. Here, utilizing the high-cadence multiwavelength monitoring on NGC 4395 harboring an intermediate-mass BH, we unveil at the intermediate-mass scale for the first time, prominent departures from the DRW process at timescales shorter than ∼hours in all three nights and bands, and plausible timescale-dependent CVs in the two longest nights of observation. Furthermore, comparing SFs of NGC 4395 to four AGNs at the supermassive scale, we suggest a new scaling relation between the timescale (τ; across nearly 3 orders of magnitude) and the BH mass (M BH): τ∝MBHγ where the exponent γ is likely ≃0.6–0.8. This exponent differs from most previous measurements, but confirms a few, and is consistent with a recent theoretical prediction, suggesting a similar accretion process in AGNs across different mass scales.

Tehran Stock Market efficiency: A quantile autoregression approach

The purpose of this paper is to evaluate the price efficiency of the Tehran Stock Market. For this aim, we used daily stock prices of 30 large companies on the Stock Exchange. In the first stage, a unit root test with the endogenous break and without a structural break was performed using augmented dickey-fuller test (ADF) tests and Phillips-perron (PP) tests. The results indicate that the price of 9 companies has a random walk process with intercept and 21 companies follow a random walk without intercept and trend component process which is known as the pure random walk process. Thus, considering the ADF and PP tests, most companies’ stock prices are efficient. Quantile autoregression results in the second stage show that the stock prices in the middle price deciles have weak efficiency, but in the lower and upper price deciles, the stock price does not follow the weak efficiency conditions. So, if the stock price deviates (up or down) from the long-term mean, the market becomes inefficient, but when the stock price is at the median level, the market is efficient. The general conclusion is that median prices are the long-term average prices that change over time, and stock prices tend to move toward that price.

TESTING MEAN REVERSION OF STOCK PRICES IN OECD COUNTRIES: EVIDENCE FROM FOURIER THRESHOLD UNIT ROOT TEST

Researchers focus on whether stock prices have a unit root, that is, whether they contain a random walk process. If stock prices have a stationary process, that is, if they return to the mean, the effects of shocks are temporary, and it is interpreted that they will return to the trend path over time. If stock prices have transitory shocks, it allows for the prediction of future movements based on past behavior in terms of investment. This study investigates whether stock prices revert to the mean and thus have a random walk process. For this purpose, the Fourier Threshold Unit Root (FTUR) test based on the test methodology of Caner and Hansen (2001) for the period January 1990–January 2021 for 26 OECD countries is applied. The FTUR test takes into account both structural breaks and nonlinearities. The purpose of using Fourier functions to account for structural changes is that they are not affected by the number, location, or shape of breaks. Thus, the power of the test increases. According to the results of this test, stock prices in Austria, Canada, Germany, Italy, New Zealand, Spain, and the UK are linear. Therefore, Fourier Augmented Dickey-Fuller (FADF) unit root analysis was performed for these countries. The FTUR test was performed in other countries. According to the results of FTUR and FADF unit root tests, stock prices are found to contain unit roots in some countries except Italy. In some countries, stock prices have a partial unit root structure. In other words, the effects of shocks are permanent, and it is concluded that future returns cannot be predicted in these countries with the random walk process.

Instability in the quantum restart problem.

Repeatedly monitored quantum walks with a rate 1/τ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus τ, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time is controlled not only by the restart time, as in classical problems, but also by the sampling time τ. We provide numerical evidence that this global minimum occurs for the τ minimizing the mean hitting time, given restarts taking place after each measurement. Last, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time τ.

Dependent Competing Failure Processes in Reliability Systems.

This paper deals with a reliability system hit by three types of shocks ranked as harmless, critical, or extreme, depending on their magnitudes, being below H1, between H1 and H2, and above H2, respectively. The system's failure is caused by a single extreme shock or by a total of N critical shocks. In addition, the system fails under occurrences of M pairs of shocks with lags less than some δ (δ-shocks) in any order. Thus, the system fails when one of the three named cumulative damages occurs first. Thus, it fails due to the competition of the three associated shock processes. We obtain a closed-form joint distribution of the time-to-failure, shock count upon failure, δ-shock count, and cumulative damage to the system on failure, to name a few. In particular, the reliability function directly follows from the marginal distribution of the failure time. In a modified system, we restrict δ-shocks to those with small lags between consecutive harmful shocks. We treat the system as a generalized random walk process and use an embellished variant of discrete operational calculus developed in our earlier work. We demonstrate analytical tractability of our formulas which are also validated, through Monte Carlo simulation.

Asymmetric influence-based superposed random walk link prediction algorithm in complex networks

Random walk-based link prediction algorithms have achieved desirable results for complex network mining, but in these algorithms, the transition probability of particles usually only considers node degrees, resulting in particles being able to randomly select adjacent nodes for random walks in an equal probability manner, to solve this problem, the asymmetric influence-based superposed random walk link prediction algorithm is proposed in this paper. This algorithm encourages particles to choose the next node at each step of the random walk process based on the asymmetric influence between nodes. To this end, we fully consider the topological information around each node and propose the asymmetric influence between nodes. Then, an adjustable parameter is applied to normalize the degree of nodes and the asymmetric influence between nodes into transition probability. Based on this, the proposed new transition probability is applied to superposed random walk process to measure the similarity between all nodes in the network. Empirical experiments are conducted on 16 real-world network datasets such as social network, ecology network, and animal network. The experimental results show that the proposed algorithm has high prediction accuracy in most network, compared with 10 benchmark indices.

Deep knowledge distillation: A self-mutual learning framework for traffic prediction

Deep knowledge distillation: A self-mutual learning framework for traffic prediction

Urban segregation on multilayered transport networks: a random walk approach

We present a novel method for analysing socio-spatial segregation in cities by considering constraints imposed by transportation networks. Using a multilayered network approach, we model the interaction probabilities of socio-economic groups with random walks and Lévy flights. This method allows for evaluation of new transport infrastructure’s impact on segregation while quantifying each network’s contribution to interaction opportunities. The proposed random walk segregation index measures the probability of individuals encountering diverse social groups based on their available means of transit via random walks. The index incorporates temporal constraints in urban mobility with a parameter, α∈[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in [0,1)$$\\end{document}, of the probability of the random walk continuing at each time step. By applying this to a toy model and conducting a sensitivity analysis, we explore how the index changes dependent on this temporal constraint. When the parameter equals zero, the measure simplifies to an isolation index. When the parameter approaches one it represents the city’s overall socio-economic distribution by mirroring the steady-state of the random walk process. Using Cuenca, Ecuador as a case study, we illustrate the method’s applicability in transportation planning as a valuable tool for policymakers, addressing spatial distribution of socio-economic groups and the connectivity of existing transport networks, thus promoting equitable interactions throughout the city.

Visual Feature Tuning Properties of Short-Latency Stimulus-Driven Ocular Position Drift Responses during Gaze Fixation.

Ocular position drifts during gaze fixation are significantly less well understood than microsaccades. We recently identified a short-latency ocular position drift response, of ∼1 min arc amplitude, that is triggered within <100 ms by visual onsets. This systematic eye movement response is feature-tuned and seems to be coordinated with a simultaneous resetting of the saccadic system by visual stimuli. However, much remains to be learned about the drift response, especially for designing better-informed neurophysiological experiments unraveling its mechanistic substrates. Here we systematically tested multiple new feature tuning properties of drift responses. Using highly precise eye tracking in three male rhesus macaque monkeys, we found that drift responses still occur for tiny foveal visual stimuli. Moreover, the responses exhibit size tuning, scaling their amplitude (both up and down) as a function of stimulus size, and they also possess a monotonically increasing contrast sensitivity curve. Importantly, short-latency drift responses still occur for small peripheral visual targets, which additionally introduce spatially directed modulations in drift trajectories toward the appearing peripheral stimuli. Drift responses also remain predominantly upward even for stimuli exclusively located in the lower visual field and even when starting gaze position is upward. When we checked the timing of drift responses, we found it was better synchronized to stimulus-induced saccadic inhibition than to stimulus onset. These results, along with a suppression of drift response amplitudes by peristimulus saccades, suggest that drift responses reflect the rapid impacts of short-latency and feature-tuned visual neural activity on final oculomotor control circuitry in the brain.