Simple Random Walk Research Articles

Frustrated random walks: A fast method to compute node distances on hypergraphs.

A hypergraph is a generalization of a graph that arises naturally when attribute-sharing among entities is considered. Compared to graphs, hypergraphs have the distinct advantage that they contain explicit communities and are more convenient to manipulate. An open problem in hypergraph research is how to accurately and efficiently calculate node distances on hypergraphs. Estimating node distances enables us to find a node's nearest neighbors, which has important applications in such areas as recommender system, targeted advertising, etc. In this paper, we propose using expected hitting times of random walks to compute hypergraph node distances. We note that simple random walks cannot accurately compute node distances on highly complex real-world hypergraphs, which motivates us to introduce frustrated random walks (FRW) for this task. We further benchmark our method against DeepWalk, and show that while the latter can achieve comparable results, FRW has a distinct computational advantage in cases where the number of targets is fairly small. For such cases, we show that FRW runs in significantly shorter time than DeepWalk. Finally, we analyze the time complexity of our method, and show that for large and sparse hypergraphs, the complexity is approximately linear, rendering it superior to the DeepWalk alternative.

Localization of a one-dimensional simple random walk among power-law renewal obstacles

Localization of a one-dimensional simple random walk among power-law renewal obstacles

Tracking and modeling the movement of Queensland fruit flies, Bactrocera tryoni, using harmonic radar in papaya fields

Determining movement parameters for pest insects such as tephritid fruit flies is critical to developing models which can be used to increase the effectiveness of surveillance and control strategies. In this study, harmonic radar was used to track wild-caught male Queensland fruit flies (Qflies), Bactrocera tryoni, in papaya fields. Experiment 1 continuously tracked single flies which were prodded to induce movement. Qfly movements from this experiment showed greater mean squared displacement than predicted by both a simple random walk (RW) or a correlated random walk (CRW) model, suggesting that movement parameters derived from the entire data set do not adequately describe the movement of individual Qfly at all spatial scales or for all behavioral states. This conclusion is supported by both fractal and hidden Markov model (HMM) analysis. Lower fractal dimensions (straighter movement paths) were observed at larger spatial scales (> 2.5 m) suggesting that Qflies have qualitatively distinct movement at different scales. Further, a two-state HMM fit the observed movement data better than the CRW or RW models. Experiment 2 identified individual landing locations, twice a day, for groups of released Qflies, demonstrating that flies could be tracked over longer periods of time.

Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk

Three Favorite Edges Occurs Infinitely Often for One-Dimensional Simple Random Walk

Transition of the Simple Random Walk on the Ice Model Graph

The 6-vertex model holds significance in various mathematical and physical domains. The configurations of the 6-vertex model correspond to the paths in multigraphs. This article focuses on calculating the transition probability for the simple random walk on these multigraphs. An intriguing aspect of the findings is the utilization of continued fractions in the computation of the transition probability.

Universal inequalities for Dirichlet eigenvalues on discrete groups

We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary condition on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Pólya, Yau, and others. Here we focus on a version by Cheng and Yang. Specifically, we prove Yang-type universal inequalities for Cayley graphs of finitely generated amenable groups, as well as for the d -regular tree (simple random walk on the free group).

Persistent and anti-persistent motion in bounded and unbounded space: resolution of the first-passage problem

The presence of temporal correlations in random movement trajectories is a widespread phenomenon across biological, chemical and physical systems. The ubiquity of persistent and anti-persistent motion in many natural and synthetic systems has led to a large literature on the modelling of temporally correlated movement paths. Despite the substantial body of work, little progress has been made to determine the dynamical properties of various transport related quantities, including the first-passage or first-hitting probability to one or multiple absorbing targets when space is bounded. To bridge this knowledge gap we generalise the renewal theory of first-passage and splitting probabilities to correlated discrete variables. We do so in arbitrary dimensions on a lattice for the so-called correlated or persistent random walk, the one step non-Markovian extension of the simple lattice random walk in bounded and unbounded space. We focus on bounded domains and consider both persistent and anti-persistent motion in hypercubic lattices as well as the hexagonal lattice. The discrete formalism allows us to extend the notion of the first-passage to that of the directional first-passage, whereby the walker must reach the target from a prescribed direction for a hitting event to occur. As an application to spatio-temporal observations of correlated moving cells that may be either repelled or attracted to hard surfaces, we compare the first-passage statistics to a target within a reflecting domain depending on whether an interaction with the reflective interface invokes a reversal of the movement direction or not. With strong persistence we observe multi-modality in the first-passage distribution in the former case, which instead is greatly suppressed in the latter.

Cover-time Gumbel fluctuations in finite-range, symmetric, irreducible random walks on torus

In this paper, we provide the mathematical foundation for an explicit and universal feature of cover time for a large class of random work processes, which was previously observed by Chupeau et al (2015 Nat. Phys. 11 844–7). Specifically, we rigorously establish that the fluctuations of the cover time, normalized by the mean first passage time, follow a Gumbel distribution, for finite-range, symmetric, irreducible random walks on a torus of dimension three or higher. The result contributes to a better understanding of cover-time behavior in random search processes, especially on the efficiency of exhaustive searches. Our approach builds upon the work of Belius (2013 Probab. Theory Relat. Fields 157 635–89) on cover times for simple random walks, leveraging a strong coupling between the random walk and random interlacements.

Dirac walks on regular trees

The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al (2022 Linear Multilinear Algebr. 72 325–65) proposed a classical ‘Dirac walk’ diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such ‘Dirac walks’ on the Bethe lattice, a d-regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ( ds=1 for d = 2 and ds=3 for d⩾3 ).

Solute transport along a single channel with radial diffusion in the porous rock matrix: A simple analytical solution and the implementation of time domain random walk algorithm

Numerous field observations show that the channels in one fracture are narrow and the solute penetration depth might be larger than the width. For this case, the diffusion from a channel into the matrix is more realistic to be modeled as radial diffusion than one-dimensional. In present work, the single channel model with radial diffusion is revisited and a simple and robust analytical solution is developed. This solution takes a convolution form of two functions, in which different transport mechanisms are accounted for. The statistical interpretations of the two functions and the analytical solution aid to develop a simple Time Domain Random Walk (TDRW) algorithm and an extension is made to improve its accuracy, efficiency and applicability. To demonstrate the accuracy and efficacy of the extended algorithm, three groups of simulations are performed and it is found that the results of all approaches are identical. The TDRW algorithm, having the same performance as that of inverse Laplace transform solution, is superior to Gaussian quadrature method in computational time. However, due to Monte Carlo nature of the algorithm, the computational burden of TDRW algorithm is dependent on the number of particles applied, which also influences the calculation accuracy. Therefore, a trade-off between computational burden and calculation accuracy should always be made, once the TDRW algorithm is used.

On Pólya’s random walk constants

A celebrated result in probability theory is that a simple symmetric random walk on the d d -dimensional lattice Z d \mathbb {Z}^d is recurrent for d = 1 , 2 d=1,2 and transient for d ≥ 3 d\geq 3 . In this note, we derive a closed-form expression, in terms of the Lauricella function F C F_C , for the return probability for all d ≥ 3 d\geq 3 . Previously, a closed-form formula had only been available for d = 3 d=3 .

Sharp estimates for distinguished random walks on affine buildings of type [formula omitted]

We study a distinguished random walk on affine buildings of type A˜r , which was already considered by Cartwright, Saloff-Coste and Woess. In rank r=2, it is the simple random walk and we obtain optimal global bounds for its transition density (same upper and lower bound, up to multiplicative constants). In the higher rank case, we obtain sharp uniform bounds in fairly large space–time regions which are sufficient for most applications.

The spectrum of asymptotic Cayley trees

We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a Cayley tree and, in general, a non-empty pure point spectrum. We apply our results to studying continuous time quantum walk on these graphs. If the pure point spectrum is nonempty the walk is in general confined with a nonzero probability.

Knowledge attitude and convenience on self-medication practices among university students in Bangladesh exploration using structural equation modeling approach

Self-medication is a prevalent practice among university students globally and is a significant public health concern. However, previous research has been limited in scope, focusing primarily on adolescents or the general public, leaving a gap in understanding the causal relationships associated with self-medication; thus, this study aimed to investigate the factors influencing self-medication practices among university students in Bangladesh by developing a comprehensive causal model. Data from 417 students across five public universities were collected using the simple random walk technique by a team of 10 members. The study utilized constructs of knowledge, attitude, and convenience related to self-medication as independent variables, while self-medication practice as the dependent variable. One-way ANOVA and structural equation modeling (SEM) were employed to develop a causal model of self-medication practice among university students in Bangladesh. The findings revealed that students with better medication knowledge and adverse drug reactions (ADRs) were more likely to practice self-medication. A positive attitude towards self-medication and ADRs was also significantly associated with higher self-medication practice scores. Additionally, those who perceived self-medication as convenient and prescribed medication as inconvenient had higher self-medication practice scores. The attitude towards self-medication had the most substantial negative effect on self-medication practice, followed by the inconvenience of prescribed medication and the convenience of self-medication. The model explained 87% of the variance in self-medication practice, indicating a good fit for the data. University students in Bangladesh possess intermediate knowledge of medication and primary knowledge of ADRs. They exhibit a positive attitude towards self-medication and ADRs. Physical convenience favors self-medication, while the inconvenience of prescribed medication contributes to its lower preference. Policymakers should focus on evidence-based guidelines to reduce the extent of unnecessary self-medication practice and to enhance the quantity and accessibility of prescribed medications to address the issue effectively.

Random walk model for dual cascades in wave turbulence.

Dual cascades in turbulent systems with two conserved quadratic quantities famously arise in both two-dimensional hydrodynamic turbulence and also in wave turbulence based on four-wave interactions. Examples for the latter include surface waves and nonlinear Schrödinger equationswith cubic nonlinearity. However, numerical simulations in forced-dissipative equilibrium of two-dimensional turbulence and of a one-dimensional wave system reveal that the physical nature of their cascades is starkly different. This is demonstrated by comparing their spectra in a finite inertial range and by comparing the temporal fluctuations of their spectral fluxes. In particular, the flux fluctuations are much larger in the wave case and frequently lead to instantaneous flux values that have the opposite sign of the mean flux, a phenomenon that is completely absent in the hydrodynamic case. A simple random walk model for the dual cascade in wave turbulence is then formulated that is very successful in explaining these effects. In particular, the model is able to replicate the detailed shape of the observed turbulent spectrum in a finite inertial range, and it also offers a ready explanation for the large flux fluctuations. It is also shown that a nonlinear diffusion model for the wave system cannot explain the observed spectral shapes. Overall, this suggests that in wave turbulence the systematic spectral fluxes observed in a dual cascade do not require an irreversible dynamical mechanism, rather, they arise as the inevitable outcome of blind chance.

Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks.

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási-Albert (BA), Erdös-Rényi (ER), and Watts-Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM<R0. For RM,R0>1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.

Weighted Jump in Random Walk graph sampling

Graph sampling is a challenging problem in network analysis due to the complex structures of networks. Currently, a series of graph sampling algorithms based on random walks achieve good results in graph sampling tasks. However, the existing methods often reduce the conductance of graphs, causing the sampler to stay in the same node for a long time. This results in undersampling. In this paper, we propose a novel Weighted Jump Random Walk (WJRW) algorithm to generate representative samples. We design a parameter in the WJRW algorithm that can adjust the proportions of random walk and random jump in every step. According to the issue of repeated sample nodes in the Generalized Maximum Degree (GMD) method and the problem of large deviations in the Simple Random Walk (SRW), WJRW addresses the weaknesses of the GMD method and enhances the diffusion of a random walker on graphs, leading to a more representative sample. Then, WJRW addresses the issue of large deviations in SRW and enhances the efficiency of the unbiased estimator. By generating smoother stationary distributions. Numerical experiments with extensive real-world networks verify that our method achieves higher accuracy than classical and state-of-the-art methods in estimating distribution estimation.

A multivariate extension of the Erdős–Taylor theorem

The Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if LN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N$$\\end{document} is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then πlogNLN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N} \\,\ extsf{L}_N$$\\end{document} converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if LN(1,2)=∑n=1N1{Sn(1)=Sn(2)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N^{(1,2)}=\\sum _{n=1}^N \\mathbb {1}_{\\{S_n^{(1)}= S_n^{(2)}\\}}$$\\end{document}, then πlogNLN(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N}\\, \ extsf{L}^{(1,2)}_N$$\\end{document} converges in distribution to an exponential random variable of parameter one. We prove that for every h⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\geqslant 3$$\\end{document}, the family {πlogNLN(i,j)}1⩽i<j⩽h\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big \\{ \\frac{\\pi }{\\log N} \\,\ extsf{L}_N^{(i,j)} \\big \\}_{1\\leqslant i<j\\leqslant h}$$\\end{document}, of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.

The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions

We show that the Brownian continuum random tree is the Gromov–Hausdorff–Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the d-dimensional torus Znd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_n^d$$\\end{document} with d>4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d>4$$\\end{document}, the hypercube {0,1}n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,1\\}^n$$\\end{document}, and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.

Mixing time of random walk on dynamical random cluster

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n2/μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^2/\\mu $$\\end{document}. In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.