Independent Random Walks Research Articles

Graph Degree Heterogeneity Facilitates Random Walker Meetings

Various graph algorithms have been developed with multiple random walks, the movement of several independent random walkers on a graph. Designing an efficient graph algorithm based on multiple random walks requires investigating multiple random walks theoretically to attain a deep understanding of their characteristics. The first meeting time is one of the important metrics for multiple random walks. The first meeting time on a graph is defined by the time it takes for multiple random walkers to meet at the same node in a graph. This time is closely related to the rendezvous problem, a fundamental problem in computer science. The first meeting time of multiple random walks has been analyzed previously, but many of these analyses have focused on regular graphs. In this paper, we analyze the first meeting time of multiple random walks in arbitrary graphs and clarify the effects of graph structures on expected values. First, we derive the spectral formula of the expected first meeting time on the basis of spectral graph theory. Then, we examine the principal component of the expected first meeting time using the derived spectral formula. The clarified principal component reveals that (a)the expected first meeting time is almost dominated by $n/(1+d_{\rm std}^2/d_{\rm avg}^2)$ and (b)the expected first meeting time is independent of the starting nodes of random walkers, where $n$ is the number of nodes of the graph. $d_{\rm avg}$ and $d_{\rm std}$ are the average and the standard deviation of weighted node degrees, respectively. The characteristic(a) is useful for understanding the effect of the graph structure on the first meeting time. According to the revealed effect of graph structures, the variance of the coefficient $d_{\rm std}/d_{\rm avg}$(degree heterogeneity) for weighted degrees facilitates the meeting of random walkers.

Simulating two-phase taxi service process by random walk theory.

City taxi service systems have been empirically studied by a number of data-driven methods. However, their underlying mechanisms are hard to understand because the present mathematical models neglect to explain a (whole) taxi service process that includes a pair of on-load phase and off-load phase. In this paper, by analyzing a large amount of taxi servicing data from a large city in China, we observe that the taxi service process shows different temporal and spatial features according to the on-load phase and off-load phase. Moreover, our correlation analysis results demonstrate the lack of dependence between the on-load phase and the off-load phase. Hence, we introduce two independent random walk models based on the Langevin equation to describe the underlying mechanism and to understand the temporal and spatial features of the taxi service process. Our study attempts to formulate the mathematical framework for simulating the taxi service process and better understanding of its underlying mechanism.

Minima of independent time-inhomogeneous random walks

In this paper, we will consider the minima of an exponentially growing number of independent time-inhomogeneous random walks, where the first- and second-order limit behaviors for the minima have been obtained.

Random Walk on the Simple Symmetric Exclusion Process

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density rho in [0, 1] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities rho except for at most two values rho _-, rho _+ in [0, 1]. The asymptotic speed we obtain in our LLN is a monotone function of rho . Also, rho _- and rho _+ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

The End Time of SIS Epidemics Driven by Random Walks on Edge-Transitive Graphs

Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet in a node. After a random recovery time, an infected agent returns to state S and can be infected again. The end of epidemic (EoE) denotes the first time where all agents are in state S, since after this moment no further infections can occur and the epidemic stops. For the case of two agents on edge-transitive graphs, we characterize EoE as a function of the network structure by relating the Laplace transform of EoE to the Laplace transform of the meeting time of two random walks. Interestingly, this analysis shows a separation between the effect of network structure and epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically in the size of the graph) under different parameter scalings, identifying regimes where EoE converges in distribution to a proper random variable or to infinity. We also highlight the impact of different graph structures on EoE, characterizing it under complete graphs, complete bipartite graphs, and rings.

RETRACTED: The impact of the presence of autoregressive conditional heteroscedasticity (ARCH) effects on spurious regressions

Spurious (nonsensical) regressions with independent random walks or even with stationary series are well known. However, how their spuriosity is affected by nonlinearity in series has been scantly addressed. In this study, we examine, using Monte Carlo analysis, the effect of autoregressive conditional Heteroskedasticity (ARCH) on nonsensical regressions and we find that ARCH can neutralize most of spuriosity. Specifically, our analysis of finite sample behavior of the t-ratio in a spurious regression framework where ARCH effects are included in a Data Generating Process (DGP) model and Monte Carlo experiments show that large ARCH effects somehow weaken the degree of spuriosity. This will have implications for unit root and cointegration analysis. Our simulations suggest that many of the regressions in the literature, based on individual predictor variables, may be spurious.

Continuity for the asymptotic shape in the frog model with random initial configurations

We consider the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. An interest of this model is how initial configurations affect the asymptotic shape of the set of all sites visited by active particles up to a certain time. Thus, in this paper, we prove continuity for the asymptotic shape in the law of the initial configuration.

The social network model on infinite graphs

Given an infinite connected regular graph $G=(V,E)$, place at each vertex $\operatorname{Poisson}(\lambda)$ walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when $G$ is vertex-transitive and amenable, for all $\lambda>0$ a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when $G$ is nonamenable (not necessarily transitive) there is always a phase transition at some $\lambda_{\mathrm{c}}(G)>0$. We give general bounds on $\lambda_{\mathrm{c}}(G)$ and study the case that $G$ is the $d$-regular tree in more detail. Finally, we show that in the nonamenable setup, for every $\lambda$ there exists a finite time $t_{\lambda}(G)$ such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time $t_{\lambda}(G)$.

A paradox for expected hitting times

We prove a counterintuitive result concerning the expected hitting/absorption time for a class of Markov chains. The “paradox” already shows itself in the following elementary example that is suitable for undergraduate teaching: Batman and the Joker perform independent discrete-time random walks on the vertices of a square until they meet, starting from opposite vertices. Batman always moves (and clockwise and anticlockwise steps are equally likely), while the Joker remains still on any given step with probability On average the Joker survives for twice as long by staying still with arbitrarily small but positive probability (in the limit as ) than by always moving (when q = 0).

Gumbel and Fréchet convergence of the maxima of independent random walks

AbstractWe consider point process convergence for sequences of independent and identically distributed random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions. The proofs depend heavily on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.

Large deviations for the maximum of a branching random walk with stretched exponential tails

We prove large deviation results for the position of the rightmost particle, denoted by $M_{n}$, in a one-dimensional branching random walk in a case when Cramér’s condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e. both tails decay as $e^{-\Theta (|t|^{r})}$ for some $r\in ( 0,1)$. It is known that in this case, $M_{n}$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_{n}$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $\tilde{M} _{n}$, and we show a large deviation principle for the laws of $n^{-1/r}\tilde{M} _{n}$ as well.

RETRACTED: The impact of the presence of autoregressive conditional heteroscedasticity (ARCH) effects on spurious regressions

Abstract Spurious (nonsensical) regressions with independent random walks or even with stationary series are well known. However, how their spuriosity is affected by nonlinearity in series has been scantly addressed. In this study, I examine, using Monte Carlo analysis, the effect of autoregressive conditional Heteroskedasticity (ARCH) on nonsensical regressions and I find that ARCH can neutralize most of spuriosity. Specifically, my analysis of finite sample behavior of the t-ratio in a spurious regression framework where ARCH effects are included in a Data Generating Process (DGP) model and Monte Carlo experiments show that large ARCH effects somehow weaken the degree of spuriosity. This will have implications for unit root and cointegration analysis. My simulations suggest that many of the regressions in the literature, based on individual predictor variables, may be spurious.

Random Walk Based Sampling for Load Balancing in Multi-Server Systems

In multi-server systems, a classical job assignment algorithm works as follows: at the arrival of each job, pick d servers independently and uniformly at random and send the job to the least loaded server among the d servers. This model is known as the powerof- d choices algorithm. In this paper, we analyze a variant of this algorithm, where d servers are sampled through d independent non-backtracking random walks on a k-regular graph. The random walkers are periodically reset to independent uniform random positions. Under some assumptions on the underlying graph, we show that the system dynamics under this new algorithm converges to the solution of a deterministic ordinary differential equation (ODE), which is the same ODE as the classical power-of-d choices. We also show that the new algorithm stablizes the system, and the stationary distribution of the system converges to the stationary solution of the ODE. The new scheme can be considered as a derandomized version of power-of-d choices as it reduces the use of randomness while maintaining the performance of power-of-d choices.

HAR Testing for Spurious Regression in Trend

The usual t test, the t test based on heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators, and the heteroskedasticity and autocorrelation robust (HAR) test are three statistics that are widely used in applied econometric work. The use of these significance tests in trend regression is of particular interest given the potential for spurious relationships in trend formulations. Following a longstanding tradition in the spurious regression literature, this paper investigates the asymptotic and finite sample properties of these test statistics in several spurious regression contexts, including regression of stochastic trends on time polynomials and regressions among independent random walks. Concordant with existing theory (Phillips 1986, 1998; Sun 2004, 2014b) the usual t test and HAC standardized test fail to control size as the sample size n → ∞ in these spurious formulations, whereas HAR tests converge to well-defined limit distributions in each case and therefore have the capacity to be consistent and control size. However, it is shown that when the number of trend regressors K → ∞ , all three statistics, including the HAR test, diverge and fail to control size as n → ∞ . These findings are relevant to high-dimensional nonstationary time series regressions where machine learning methods may be employed.

Decomposing correlated random walks on common and counter movements

Random walk is one of the most classical models in probability theory which has got extensive applications in many areas and is still of great interest in practice. For those problems that require modelling two random walks on lattice, correlation of the random walks is non-ignorable. This paper presents a new method to study the dependency structure of two generally correlated random walks. By introducing a change-of-time process, two correlated random walks can be decomposed into sum/difference of two independent random walks with time change, where the two independent random walks present respectively the common movements and counter movements of the original random walks. A sufficient and necessary condition is given for the mutual independence of the change-of-time process and the two independent random walks. For the prospective applications of the decomposition method in theory and practice, we consider the calculations of the characteristic functions for Markovian and non-Markovian random walks and an empirical example in futures trading is given.

Universality of local statistics for noncolliding random walks

We consider the $N$-particle noncolliding Bernoulli random walk—a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is in the “bulk”), and the initial configuration is balanced in a certain sense, then the space-time local statistics at $x$ are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk. The noncolliding Bernoulli random walk is a discrete analogue of the $\boldsymbol{\beta}=2$ Dyson Brownian motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale $T$, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior.

Parking on transitive unimodular graphs

Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p\geq1/2$, and only finitely many times otherwise.

Uphill migration in coupled driven particle systems

In particle systems subject to a nonuniform drive, particle migration is observed from the driven to the non-driven region and vice-versa, depending on details of the hopping dynamics, leading to apparent violations of Fick’s law and of steady-state thermodynamics. We propose and discuss a very basic model in the framework of independent random walkers on a pair of rings, one of which features biased hopping rates, in which this phenomenon is observed and fully explained.

Transport in Quantum Multi-barrier Systems as Random Walks on a Lattice

A quantum finite multi-barrier system, with a periodic potential, is considered and exact expressions for its plane wave amplitudes are obtained using the Transfer Matrix method (Colangeli et al. in J Stat Mech Theor Exp 6:P06006, 2015). This quantum model is then associated with a stochastic process of independent random walks on a lattice, by properly relating the wave amplitudes with the hopping probabilities of the particles moving on the lattice and with the injection rates from external particle reservoirs. Analytical and numerical results prove that the stationary density profile of the particle system overlaps with the quantum mass density profile of the stationary Schrödinger equation, when the parameters of the two models are suitably matched. The equivalence between the quantum model and a stochastic particle system would mainly be fruitful in a disordered setup. Indeed, we also show, here, that this connection, analytically proven to hold for periodic barriers, holds even when the width of the barriers and the distance between barriers are randomly chosen.

Gossip algorithm for multi-agent systems via random walk

This paper proposes an asynchronous gossip framework where agents move according to independent random walks over a location graph and interactions may occur only when two agents share the same location. Our goal is to investigate how average consensus may be achieved when agents’ motion occurs over a set of discrete locations with topological constraints. This could be used to model the spreading of information across moving crowds or the coordination of agents monitoring a discrete set of points of interest.