Random Walk Research Articles

Effects of Predation-Induced Emigration on a Landscape Ecological Model

Predators impact prey populations directly through consumption and indirectly via trait-mediated effects like predator-induced emigration (PIE), where prey alter movement due to predation risk. While PIE can significantly influence prey dynamics, its combined effect with direct predation in fragmented habitats is underexplored. Habitat fragmentation reduces viable habitats and isolates populations, necessitating an understanding of these interactions for conservation. In this paper, we present a reaction–diffusion model to investigate prey persistence under both direct predation and PIE in fragmented landscapes. The model considers prey growing logistically within a bounded habitat patch surrounded by a hostile matrix. Prey move via unbiased random walks internally but exhibit biased movement at habitat boundaries influenced by predation risk. Predators are assumed constant, operating on a different timescale. We examine three predation functional responses—constant yield, Holling Type I, and Holling Type III—and three emigration patterns: density-independent, positive density-dependent, and negative density-dependent emigration. Using the method of sub- and supersolutions, we establish conditions for the existence and multiplicity of positive steady-state solutions. Numerical simulations in one-dimensional habitats further elucidate the structure of these solutions. Our findings demonstrate that the interplay between direct predation and PIE crucially affects prey persistence in fragmented habitats. Depending on the functional response and emigration pattern, PIE can either mitigate or amplify the impact of direct predation. This underscores the importance of incorporating both direct and indirect predation effects in ecological models to better predict species dynamics and inform conservation strategies in fragmented landscapes.

Comparative Analysis of Human Mortality Models: Insights from Denmark and the Netherlands

This study provides a comparative analysis of human mortality models for Denmark and the Netherlands from 1970 to 2008, using data from the Human Mortality Database (HMD). The analysis explores mortality trends across age groups and genders, highlighting key patterns and differences between the two countries. The Lee-Carter model is applied to analyze mortality dynamics, while the Gompertz model estimates age-specific mortality rates, both proving effective in capturing long-term trends. The study finds consistent differences between male and female mortality, with men experiencing higher mortality rates across all age groups in both countries. Mortality rates have declined significantly over time due to advancements in healthcare, socioeconomic improvements, and better living conditions. To project future mortality trends, the Random Walk with Drift model is used, offering insights into longevity and life expectancy for both populations. The results emphasize the need for ongoing public health improvements and policies to address the challenges posed by aging populations

Recurrences Reveal Shared Causal Drivers of Complex Time Series

Unmeasured causal forces influence diverse experimental time series, such as the transcription factors that regulate genes or the descending neurons that steer motor circuits. Combining the theory of skew-product dynamical systems with topological data analysis, we show that simultaneous recurrence events across multiple time series reveal the structure of their shared unobserved driving signal. We introduce a physics-based unsupervised learning algorithm that reconstructs causal drivers by iteratively building a recurrence graph with glasslike structure. As the amount of data increases, a percolation transition on this graph leads to weak ergodicity breaking for random walks—revealing the shared driver’s dynamics, even from strongly corrupted measurements. We relate reconstruction accuracy to the rate of information transfer from a chaotic driver to the response systems, and we find that effective reconstruction proceeds through gradual approximation of the driver’s dynamical attractor. Through extensive benchmarks against classical signal processing and machine learning techniques, we demonstrate our method’s ability to extract causal drivers from diverse experimental datasets spanning ecology, genomics, fluid dynamics, and physiology. Published by the American Physical Society 2025

Semi-infinite particle systems with exclusion interaction and heterogeneous jump rates

Abstract We study semi-infinite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. Under appropriate hypotheses on the jump rates (uniformly bounded rates is sufficient) and started from an initial condition that is a finite perturbation of the close-packed configuration, we give conditions under which the particles evolve as a single, semi-infinite “stable cloud”. More precisely, we show that inter-particle separations converge to a product-geometric stationary distribution, and that the location of every particle obeys a strong law of large numbers with the same characteristic speed.

First Passage Percolation, Local Uniqueness for Interlacements and Capacity of Random Walk

The study of first passage percolation (FPP) for the random interlacements model has been initiated in Andres and Prévost (Ann Appl Probab 34(2):1846–1895), where it is shown that on Zd, d≥3, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball.

Random walk with horizontal and cyclic currents

Random walk with horizontal and cyclic currents

Link prediction of heterogeneous complex networks based on an improved embedding learning algorithm.

Link prediction in heterogeneous networks is an active research topic in the field of complex network science. Recognizing the limitations of existing methods, which often overlook the varying contributions of different local structures within these networks, this study introduces a novel algorithm named SW-Metapath2vec. This algorithm enhances the embedding learning process by assigning weights to meta-path traces generated through random walks and translates the potential connections between nodes into the cosine similarity of embedded vectors. The study was conducted using multiple real-world and synthetic datasets to validate the proposed algorithm's performance. The results indicate that SW-Metapath2vec significantly outperforms benchmark algorithms. Notably, the algorithm maintains high predictive performance even when a substantial proportion of network nodes are removed, demonstrating its resilience and potential for practical application in analyzing large-scale heterogeneous networks. These findings contribute to the advancement of link prediction techniques and offer valuable insights and tools for related research areas.

Sound and Complete Proof Rules for Probabilistic Termination

Deciding termination is a fundamental problem in the analysis of probabilistic imperative programs. We consider the qualitative and quantitative probabilistic termination problems for an imperative programming model with discrete probabilistic choice and demonic bounded nondeterminism. The qualitative question asks if the program terminates almost-surely, no matter how nondeterminism is resolved. The quantitative question asks for a bound on the probability of termination. Despite a long and rich literature on the topic, no sound and relatively complete proof systems were known for these problems. In this paper, we provide sound and relatively complete proof rules for proving qualitative and quantitative termination in the assertion language of arithmetic. Our rules use variant functions as measures of distances to termination as well as supermartingales that restrain the growth of these variants almost-surely. Our completeness result shows how to construct suitable supermartingale and variant functions from an almost-surely terminating program. We also show that proofs of termination in many existing proof systems can be transformed to proofs in our system, pointing to its applicability in practice. As an application of our proof rule, we show an explicit proof of almost-sure termination for the two-dimensional random walker.

Evolving Motility of Active Droplets is Captured by a Self-Repelling Random Walk Model

Evolving Motility of Active Droplets is Captured by a Self-Repelling Random Walk Model

Weak dependence and optimal quantitative self-normalized central limit theorems

Consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry–Esseen bound of optimal order n^{-1/2} for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and L^{p} ) distances. The results show that in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate n^{-1/2} cannot be reached, refuting a popular belief in the literature. This can be salvaged by simple methods. We reveal that in order to maintain the optimal speed of convergence n^{-1/2} , simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of GARCH models of any order, functionals of dynamical systems arising from stochastic differential equations, iterated random functions and many more.

Random walks with long-range memory on networks.

We study an exactly solvable random walk model with long-range memory on arbitrary networks. The walker performs unbiased random steps to nearest-neighbor nodes and intermittently resets to previously visited nodes in a preferential way such that the most visited nodes have proportionally a higher probability to be chosen for revisit. The occupation probability can be expressed as a sum over the eigenmodes of the standard random walk matrix of the network, where the amplitudes slowly decay as power-laws at large times, instead of exponentially. The stationary state is the same as in the absence of memory, and detailed balance is fulfilled. However, the relaxation of the transient part becomes critically self-organized at late times, as it is dominated by a single power-law whose exponent depends on the second largest eigenvalue and on the resetting probability. We apply our findings to finite networks, such as rings, complete graphs, Watts-Strogatz, and Barabási-Albert networks, and to Barbell and comb-like graphs. Our study could be of interest for modeling complex transport phenomena, such as human mobility, epidemic spreading, or animal foraging.

Optimizing cost through dynamic stochastic resetting

Abstract The cost of stochastic resetting is considered within the context of a discrete random walk (RW) model. In addition to standard stochastic resetting, for which a reset occurs with a certain probability after each step, we introduce a novel resetting protocol which we dubbed dynamic resetting. This protocol entails an additional dynamic constraint related to the direction of successive steps of the RW. We study this novel protocol for a one-dimensional RW on an infinite lattice. We analyze the impact of the constraint on the walker’s mean-first passage time and the cost (fluctuations) of the resets as a function of distance of target from the resetting location. Further, cost optimized search strategies are discussed.

Random walk models in the life sciences: including births, deaths and local interactions.

Random walks and related spatial stochastic models have been used in a range of application areas, including animal and plant ecology, infectious disease epidemiology, developmental biology, wound healing and oncology. Classical random walk models assume that all individuals in a population behave independently, ignoring local physical and biological interactions. This assumption simplifies the mathematical description of the population considerably, enabling continuum-limit descriptions to be derived and used in model analysis and fitting. However, interactions between individuals can have a crucial impact on population-level behaviour. In recent decades, research has increasingly been directed towards models that include interactions, including physical crowding effects and local biological processes such as adhesion, competition, dispersal, predation and adaptive directional bias. In this article, we review the progress that has been made with models of interacting individuals. We aim to provide an overview that is accessible to researchers in application areas, as well as to specialist modellers. We focus particularly on derivation of asymptotically exact or approximate continuum-limit descriptions and simplified deterministic models of mean-field behaviour and resulting spatial patterns. We provide worked examples and illustrative results of selected models. We conclude with a discussion of current areas of focus and future challenges.

Random walks on scale-free flowers with stochastic resetting.

This study explores the impact of stochastic resetting on the random walk dynamics within scale-free (u,v)-flowers. Utilizing the generating function technique, we develop a recursive relationship for the generating function of the first passage time and establish a connection between the mean first passage time with and without resetting. Our investigation spans multiple scenarios, with the random walker starting from various positions and aiming to reach different target nodes, allowing us to identify the optimal resetting probability that minimizes the mean first passage time for each case. We demonstrate that stochastic resetting significantly improves search efficiency, especially in larger networks. These findings underscore the effectiveness of stochastic resetting as a strategy for optimizing search algorithms in complex networks, offering valuable applications in domains such as biological transport, data networks, and search processes where rapid and efficient exploration is vital.

Random walks with stochastic resetting in complex networks: A discrete-time approach.

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time probability density functions, and in the light-tailed case, we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix, which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barabási-Albert random graphs. We show nontrivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes, and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.

Random walk models of advection-diffusion in layered media

Random walk models of advection-diffusion in layered media

Recurrence and transience of the critical random walk snake in random conductances

Recurrence and transience of the critical random walk snake in random conductances

A classification of asymptotic behaviors of Green functions of random walks in the quadrant

A classification of asymptotic behaviors of Green functions of random walks in the quadrant

FILTERING STRATEGY TO MINIMIZE ERRORS IN THE ESTIMATION OF THE Z ANGLE (YAW) OF A ROTATING BODY WITH TWO INDEPENDENT MEMS GYROSCOPES

A MEMS gyroscope is a low-cost electronic device used to determine the angular rotation rate ? of a rigid body with respect to the its x, y, and z axes, the latter known as yaw. The measurement of the angular rotation rate on each axis is perturbed by phenomena known as bias, bias instability, and angle random walk. It is desirable to accurately calculate the yaw angle; however, the phenomena of bias, bias instability, and angle random walk make this calculation impossible; several algorithms have been reported in the literature to achieve such goal, such as the Kalman and the complementary filters. We propose an improvement in the estimation of the yaw angle of a rigid body in rotation; it is achieved through a complementary filter genetically optimized that fuses angular rotation rate readings from two independent gyroscopes. The genetic algorithm is employed to calculate the parameters of the complementary filter, operating on simulated runs of the gyroscopes. The evaluation of the resulting yaw angle precision with the calculated filter demonstrated a reduction in the absolute error of the yaw angle estimation with respect to given reference signals to up to one order or magnitude with respect to the calculation given independently by each gyroscope. The results obtained were proven empirically. Key Words: Complementary filtering, Gyroscope, Yaw angle, Allan variance, Angular velocity drift.

Information-Oriented Random Walks and Pipeline Optimization for Distributed Graph Embedding

Information-Oriented Random Walks and Pipeline Optimization for Distributed Graph Embedding