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The Markov chain Monte Carlo (MCMC) method is widely used in various fields as a powerful numerical integration technique for systems with many degrees of freedom. In MCMC methods, probabilistic state transitions can be considered as a random walk in state space, and random walks allow for sampling from complex distributions. However, paradoxically, it is necessary to carefully suppress the randomness of the random walk to improve computational efficiency. By breaking detailed balance, we can create a probability flow in the state space and perform more efficient sampling along this flow. Motivated by this idea, practical and efficient nonreversible MCMC methods have been developed over the past ten years. In particular, the lifting technique, which introduces probability flows in an extended state space, has been applied to various systems and has proven more efficient than conventional reversible updates. We review and discuss several practical approaches to implementing nonreversible MCMC methods, including the shift method in the cumulative distribution and the directed-worm algorithm. [ABSTRACT FROM AUTHOR]

Eaton (1992) considered a general parametric statistical model paired with an improper prior distribution for the parameter and proved that if a certain Markov chain, constructed using the model and the prior, is recurrent, then the improper prior is strongly admissible , which (roughly speaking) means that the generalized Bayes estimators derived from the corresponding posterior distribution are admissible. Hobert and Robert (1999) proved that Eaton's Markov chain is recurrent if and only if its so-called conjugate Markov chain is recurrent. The focus of this paper is a family of Markov chains that contains all of the conjugate chains that arise in the context of a Poisson model paired with an arbitrary improper prior for the mean parameter. Sufficient conditions for recurrence and transience are developed and these are used to establish new results concerning the strong admissibility of non-conjugate improper priors for the Poisson mean. [ABSTRACT FROM AUTHOR]

The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of selforganization because energy is a local property of configurations. For nonequilibrium systems, there is no such property for which an analogous principle holds, hence no common explanation of the diverse forms of self-organization they exhibit. However, a flurry of recent empirical results has shown that a local property of configurations called "rattling" predicts the steady states of some nonequilibrium systems, leading to claims of a far-reaching principle of nonequilibrium self-organization. But for which nonequilibrium systems is rattling accurate, and why? Wedevelop a theory of rattling in terms of Markov processes that gives simple and precise answers to these key questions. Our results show that rattling predicts a broader class of nonequilibrium steady states than has been claimed and for different reasons than have been suggested. Its predictions hold to an extent determined by the relative variance of, and correlation between, the local and global "parts" of a steady state. We show how these quantities characterize the local-global relationships of various random walks on random graphs, spin-glass dynamics, and models of animal collective behavior. Surprisingly, we find that the core idea of rattling is so general as to apply to equilibrium and nonequilibrium systems alike. [ABSTRACT FROM AUTHOR]

In this paper we look at the families of random walks arising from FI-graphs. One may think of these objects as families of nested graphs, each equipped with a natural action by a symmetric group S n , such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [9] , while a systematic study of random walks on these families were considered in [10]. In the present work, we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis [1]. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property. [ABSTRACT FROM AUTHOR]

Researchers have become interested in security games in recent decades as a result of its successful application in real-world security issues. The security model is based on the Stackelberg Security Game (SSG), in which defenders (leaders) select a defensive strategy based on the optimal reaction of attackers (followers), who, at equilibrium, select the predicted assaulting strategy as a response. These applications, on the other hand, do not account for the time constraints posed by the game's players' journey time. Furthermore, players should be able to cope with dynamic settings in which their knowledge of the environment changes on a regular basis, allowing them to perform more effectively. This research proposes a security model based on a continuous-time Reinforcement Learning (RL) approach implemented using a temporal difference method that takes prior information into account to address these issues. We use a controlled, ergodic continuous-time Markov game to model the SSG. The game framework model assumes that all information is available. We calculate the number of transitions over a time interval divided by the entire value of the holding time to estimate the transition rates. The arithmetic mean of the observed cost of the individual players is used to estimate the cost for defenders and attackers. An iterated proximal/gradient approach is used to calculate the SSG equilibrium point. We offer a continuous-time random walk method for game implementation. In a numerical case relevant to rain-forest hazards, we analyze the performance of the suggested RL security solution and discuss the problems that should be considered in future. [ABSTRACT FROM AUTHOR]

In movement analysis, correlated random walk (CRW) models often use so‐called turning angles, which are measured relative to the previous movement direction. To segregate between different movement modes, hidden Markov models (HMMs) describe movements as piecewise stationary CRWs in which the distributions of turning angles and step sizes depend on the underlying state. This typically allows for the segregation of movement modes that show different movement speeds. We show that in some cases, it may be interesting to investigate absolute angles, that is, biased random walks (BRWs) instead of turning angles. In particular, while discrimination between states in the turning angle setting can only rely on movement speed, models with absolute angles can be used to discriminate between sections of different movement directions. A preprocessing algorithm is provided that enables the analysis of absolute angles in the existing R package moveHMM. In a data set of movements of cell organelles, models using not the turning angle but the absolute angle could capture interesting additional properties. Goodness‐of‐fit was increased for HMMs with absolute angles, and HMMs with absolute angles tended to choose a higher number of states, suggesting the existence and relevance of prominent directional changes in the present data set. These results suggest that models with absolute angles can provide important information in the analysis of movement patterns if the existence and frequency of directional changes is of biological importance. [ABSTRACT FROM AUTHOR]

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 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 n 2 / μ . 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. [ABSTRACT FROM AUTHOR]

Let Q be a probability measure on a finite group G, and let H be a subgroup of G. We show that a necessary and sufficient condition for the random walk driven by Q on G to induce a Markov chain on the double coset space H\backslash G/H is that Q(gH) is constant as g ranges over any double coset of H in G. We obtain this result as a corollary of a more general theorem on the double cosets H \backslash G / K for K an arbitrary subgroup of G. As an application we study a variation on the r-top to random shuffle which we show induces an irreducible, recurrent, reversible and ergodic Markov chain on the double cosets of \mathrm {Sym}_r \times \mathrm {Sym}_{n-r} in \mathrm {Sym}_n. The transition matrix of the induced walk has remarkable spectral properties: we find its invariant distribution and its eigenvalues and hence determine its rate of convergence. [ABSTRACT FROM AUTHOR]

We show that the total variation mixing time is not quasi-isometry invariant, even for Cayley graphs. Namely, we construct a sequence of pairs of Cayley graphs with maps between them that twist the metric in a bounded way, while the ratio of the two mixing times goes to infinity. The Cayley graphs serving as an example have unbounded degrees. For non-transitive graphs, we construct bounded degree graphs for which the mixing time from the worst starting point for one graph is asymptotically smaller than the mixing time from the best starting point of the random walk on a network obtained by increasing some of the edge weights from 1 to $1+o(1)$. [ABSTRACT FROM AUTHOR]

Quantum Markov chains (QMCs) are positive maps on a trace-class space describing open quantum dynamics on graphs. Such objects have a statistical resemblance with classical random walks, while at the same time they allow for internal (quantum) degrees of freedom. In this work we study continuous-time QMCs on the integer line, half-line and finite segments, so that we are able to obtain exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures. The methods employed here are applicable to a wide range of settings, but we will restrict ourselves to classes of examples for which the Lindblad generators are induced by a single positive map, and such that the Stieltjes transforms of the measures and their inverses can be calculated explicitly. [ABSTRACT FROM AUTHOR]

Let d be a positive integer, and let A be a set in Z d that contains finitely many points with integer coordinates. We consider a standard random walk X perturbed on the set A. This means that X is a Markov chain whose transition probabilities from the points outside A coincide with those of a standard random walk on Z d , whereas the transition probabilities from the points inside A are different. We investigate the impact of the perturbation on a scaling limit of X. It turns out that if d ⩾ 2, then in a typical situation the scaling limit of X coincides with that of the underlying standard random walk. This is unlike the case d = 1, in which the scaling limit of X is usually a skew Brownian motion, a skew stable Lévy process, or some other "skew" process. The distinction between the one-dimensional and multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in Z d for d ⩾ 3. More interestingly, in the situation where the standard random walk in Z 2 is recurrent, the preservation of its Donsker scaling limit is secured by the fact that the number of visits of X to the set A is much smaller than in the one-dimensional case. As a consequence, the influence of the perturbation vanishes upon the scaling. On the other edge of the spectrum, we have the situation in which the standard random walk admits a Donsker's scaling limit, whereas its locally perturbed version does not because of huge jumps from the set A, which occur early enough. [ABSTRACT FROM AUTHOR]

This paper deals with the ruin problem of an insurance company investing its capital reserve in a risky asset with the price dynamics given by a conditional geometric Brownian motion whose parameters depend on a Markov process describing random variations in the economic and financial environments. We prove a sufficient condition on the distribution of jumps of the business process ensuring the smoothness of the ruin probability as a function of the initial capital and obtain for this function an integro-differential equation. [ABSTRACT FROM AUTHOR]

Unravelling the dynamics of network vertices is pivotal, and traditional centrality measures have limitations in adapting to structural changes, directed and weighted networks, and temporal analyses. This paper introduces a ground breaking approach ‐ hitting time‐based centrality. Utilizing network matrix notations and a random walk model on a connected network G$$ G $$, we establish a Markov chain to quantify the hitting time, hitting distance, and hitting centrality, providing a nuanced measure prioritizing central vertices. Through extensive experiments using Kendall's tau coefficient, the paper evaluates the method's correlation with actual influence in the Susceptible‐Infectious (SI) model, showcasing superior performance across diverse network sizes and structures. The hitting centrality method exhibits sensitivity to connectivity dynamics, effective incorporation of temporal dynamics, and robust handling of weighted and directed networks. Positive Kendall's tau coefficients underline the method's proficiency in prioritizing influential vertices by correlating hitting centrality values with actual infection ability. The demonstrated robustness to structural changes enhances its utility for dynamic network analysis. In conclusion, our hitting time‐based centrality approach emerges as a promising method, mitigating the shortcomings of traditional measures. By integrating information propagation speed, accommodating network dynamics, and enabling time‐dependent analyses, it offers a comprehensive tool for evaluating vertex importance and influence in complex networks. [ABSTRACT FROM AUTHOR]

A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Sibuya distributions and Yule–Simon distributions are discussed. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and random walks with disasters. [ABSTRACT FROM AUTHOR]

We provide a comparison between marginal likelihood and data augmented Markov chain Monte Carlo (MCMC) algorithms for Bayesian estimation of hidden Markov models. In particular, we focus on a specification of these models based on a multivariate Gaussian distribution for the response variables. We evaluate the performance of both methods on the basis of simulated samples and an empirical application, where we compare these approaches in terms of standard measures of estimation and time efficiency. The results show that the augmented algorithm is preferable, obtaining optimal convergence properties and producing larger values of the effective sample size. Instead of using variable dimensional techniques for model selection, we suggest a method based on the so-called parallel sampling for this aim. [ABSTRACT FROM AUTHOR]

We discuss the limit distribution of open quantum walks on the periodic graphs, particularly on the cycles. We show that under certain hypothesis, we can benefit from the theory of the classical Markov chains. Thereby we can show that under certain condition the stationary distribution is unique. For certain models, we show directly the stationary distribution. We also notice that the open quantum walks cannot be always modeled as classical Markov chains by showing that it can break some classical probability rule. By providing with some examples, we show that there can be multiple stationary states for the open quantum walks on the cycles. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method of route optimization of vacant taxicab based on the abstract grid network. Firstly, an abstract grid network is built by the quadtree technology with the vacant trajectory data embedded. Secondly, the Markov decision process framework considering the sequential dependence is used to model the problem. Finally, we solve this problem by the policy iteration and compare the method by taking the GPS trajectory data in different hours of 797 taxicabs in a typical working day in Shenzhen, comparing that with the optimal strategy without considering sequential dependence, MNP, local hotspot and random walk algorithm by simulation, respectively. The results show that: The use of quadtree can reduce the sparseness of matching probability from 80.46% to 30.18%. The average revenue per unit distance has increased by 6.10%, 8.75%, 33.24% and 60.06%. The vacant driving rate has decreased by 5.33%, 6.15%, 18.86%, and 23.18%. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices. For a given integer k such that 1 ≤ k ≤ n , the k -token graph F k (G) of G is defined as the graph whose vertices are the k -subsets of the vertex set of G , and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this article, we study the structural and spectral properties of token graphs. We describe the adjacency matrix and the Laplacian matrix of F k (G) and obtain bounds on the adjacency and Laplacian spectral radii of F k (G). Interestingly, it is found that S n , the star graph on n vertices, has the same Laplacian spectral radius as that of F k (S n). It was conjectured that for any graph G , the algebraic connectivity of F k (G) is equal to the algebraic connectivity of G. This result turned out to be a theorem, as it was proved by using the theory of the continuous Markov chain of random walks and the interchange process. However, proving this theorem using algebraic and combinatorial methods is still an open and interesting problem. Using combinatorial techniques, we prove that the theorem holds for a class of graphs that have a cut-vertex of degree n − 1. We also prove it by restricting the smallest degree of the k -token graph of G. [ABSTRACT FROM AUTHOR]

In the Fastest Mixing Markov Chain problem, we are given a graph G = (V , E) and desire the discrete-time Markov chain with smallest mixing time τ subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time τ RW of the lazy random walk on G is characterised by the edge conductance Φ of G via Cheeger's inequality: Φ - 1 ≲ τ RW ≲ Φ - 2 log | V | . Analogously, we characterise the fastest mixing time τ ⋆ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance Ψ of G: Ψ - 1 ≲ τ ⋆ ≲ Ψ - 2 (log | V |) 2 . This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only ε -close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time τ ≲ ε - 1 (diam G) 2 log | V | . Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains. [ABSTRACT FROM AUTHOR]

Suppose that a sequence of random variables is a homogeneous irreducible Markov chain with finite set of states. Let , , be random variables defined on the chain transitions. The renewal function where and , , is introduced. It is shown that this function converges to its limit at an exponential rate, and an explicit description of the exponent is given. [ABSTRACT FROM AUTHOR]

A fundamental way to analyze complex multidimensional stochastic dynamics is to describe it as diffusion on a free energy landscape—free energy as a function of reaction coordinates (RCs). For such a description to be quantitatively accurate, the RC should be chosen in an optimal way. The committor function is a primary example of an optimal RC for the description of equilibrium reaction dynamics between two states. Here, additive eigenvectors (addevs) are considered as optimal RCs to address the limitations of the committor. An addev master equation for a Markov chain is derived. A stationary solution of the equation describes a sub-ensemble of trajectories conditioned on having the same optimal RC for the forward and time-reversed dynamics in the sub-ensemble. A collection of such sub-ensembles of trajectories, called stochastic eigenmodes, can be used to describe/approximate the stochastic dynamics. A non-stationary solution describes the evolution of the probability distribution. However, in contrast to the standard master equation, it provides a time-reversible description of stochastic dynamics. It can be integrated forward and backward in time. The developed framework is illustrated on two model systems—unidirectional random walk and diffusion. [ABSTRACT FROM AUTHOR]

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension d involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing b ; (ii) discrete-time random walks in continuous space with a small time-step τ. One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits b → 0 and τ → 0 respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits b → 0 and τ → 0 for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current. [ABSTRACT FROM AUTHOR]

Let $ (X_n,Y_n) $ (X n , Y n) be a two-dimensional discrete-time controlled Markov chain. The problem of optimally controlling the chain until the sum $ X_n+Y_n $ X n + Y n is equal to either of two constants is considered. The aim is to leave the continuation region as soon as possible. The optimal control is found in two particular cases by solving the partial difference equation satisfied by the value function. [ABSTRACT FROM AUTHOR]

In this paper we consider a generalization of the coupon collector problem where we assume that the set of available coupons consists of standard coupons and an additional penalty coupon, which does not belong to the collection and interferes with collecting standard coupons. Applying Markov chain approach the following problem is solved: how many coupons (on average) one has to purchase in order to complete a collection without interference or to collect n more penalty coupons than standard coupons. Also, we obtain additional results related to the distribution of the waiting time until the collection is sampled without interference or until n more penalty coupons than standard coupons is sampled. [ABSTRACT FROM AUTHOR]

Copyright of Informatica (03505596) is the property of Slovene Society Informatika and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The main objective of this paper is to analyse investment returns using a stochastic model and inform investors about the best stock market to invest in. To this effect, a Markov chain random walk model was successfully developed and implemented on 450 monthly market returns data spanning from January 1976 to December 2020 for Canada, India, Mexico, South Africa, and Switzerland obtained from the Federal Reserves of the Bank of St. Louis. The limiting state probabilities and six-month moving crush probabilities were estimated for each country, and these were used to assess the performance of the markets. The Mexican market was observed to have the least probabilities for all the negative states, while the Indian market recorded the largest limiting probabilities. In the case of positive states, the Mexican market recorded the highest limiting probabilities, while the Indian market recorded the lowest limiting probabilities. The results showed that the Mexican market performed better than the others over the study period, whilst India performed poorly. These findings provide crucial information for market regulators and investors in setting regulations and decision-making in investment. [ABSTRACT FROM AUTHOR]

We consider a continuous-time homogeneous Markov process on the state space Z+ = {0, 1, 2, ... }. The process is interpreted as the motion of a particle. A particle may transit only to neighboring points Z+, i.e., for each single motion of the particle, its coordinate changes by 1. The process is equipped with a branching mechanism. Branching sources may be located at each point of Z+. At a moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) by the same rules as the initial particle. To such a branching Markov process there corresponds a Jacobi matrix. In terms of orthogonal polynomials corresponding to this matrix, we obtain formulas for the mean number of particles at an arbitrary fixed point of Z+ at time t > 0. The results obtained are applied to some concrete models, an exact value for the mean number of particles in terms of special functions is given, and an asymptotic formula for this quantity for large time is found. [ABSTRACT FROM AUTHOR]

Finite Markov chains, memoryless random walks on complex networks, appear commonly as models for stochastic dynamics in condensed matter physics, biophysics, ecology, epidemiology, economics, and elsewhere. Here, we review exact numerical methods for the analysis of arbitrary discrete- and continuous-time Markovian networks. We focus on numerically stable methods that are required to treat nearly reducible Markov chains, which exhibit a separation of characteristic timescales and are therefore ill-conditioned. In this metastable regime, dense linear algebra methods are afflicted by propagation of error in the finite precision arithmetic, and the kinetic Monte Carlo algorithm to simulate paths is unfeasibly inefficient. Furthermore, iterative eigendecomposition methods fail to converge without the use of nontrivial and system-specific preconditioning techniques. An alternative approach is provided by state reduction procedures, which do not require additional a priori knowledge of the Markov chain. Macroscopic dynamical quantities, such as moments of the first passage time distribution for a transition to an absorbing state, and microscopic properties, such as the stationary, committor, and visitation probabilities for nodes, can be computed robustly using state reduction algorithms. The related kinetic path sampling algorithm allows for efficient sampling of trajectories on a nearly reducible Markov chain. Thus, all of the information required to determine the kinetically relevant transition mechanisms, and to identify the states that have a dominant effect on the global dynamics, can be computed reliably even for computationally challenging models. Rare events are a ubiquitous feature of realistic dynamical systems, and so the methods described herein are valuable in many practical applications. [ABSTRACT FROM AUTHOR]

We study a natural random walk on the n × n uni-upper triangular matrices, with entries in Z / m Z , generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this random walk is O (m 2 n log n + n 2 m o (1)) . This answers a question of Stong and of Arias-Castro, Diaconis, and Stanley. [ABSTRACT FROM AUTHOR]

This paper discusses random walks on edge‐changing dynamic graphs. We prove general and improved bounds for mixing, hitting, and cover times for a random walk according to a sequence of irreducible and reversible transition matrices with the time‐invariant stationary distribution. An interesting consequence is the tight bounds of the lazy Metropolis walk on any dynamic connected graph. We also prove bounds for multiple random walks on dynamic graphs. Our results extend previous upper bounds for simple random walks on dynamic graphs and give improved and tight upper bounds in several cases. Our results reinforce the observation that time‐inhomogeneous Markov chains with an invariant stationary distribution behave almost identically to a time‐homogeneous chain. [ABSTRACT FROM AUTHOR]

Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically. This tutorial provides an in-depth introduction to Markov chains and explores their connection to graphs and random walks. We use tools from linear algebra and graph theory to describe the transition matrices of different types of Markov chains, with a particular focus on exploring properties of the eigenvalues and eigenvectors corresponding to these matrices. The results presented are relevant to a number of methods in machine learning and data mining, which we describe at various stages. Rather than being a novel academic study in its own right, this text presents a collection of known results, together with some new concepts. Moreover, the tutorial focuses on offering intuition to readers rather than formal understanding and only assumes basic exposure to concepts from linear algebra and probability theory. It is therefore accessible to students and researchers from a wide variety of disciplines. [ABSTRACT FROM AUTHOR]

The colonization of suitable yet unoccupied habitat due to natural dispersal or human introduction can benefit recovery of threatened species. Predicting habitat suitability and conflict potential of colonization areas can facilitate conservation planning.Planning for reintroduction of gray wolves (Canis lupus) to the US state of Colorado is underway. Assessing which occupancy sites minimize the likelihood of human–wolf conflict during dispersal events and seasonal movements is critical to the success of this initiative.We used a spatial absorbing Markov chain (SAMC) framework, which extends random walk theory and probabilistically accounts for both movement behaviour and mortality risk, to compare the viability of potential occupancy sites (public lands >500 km2 to minimally meet wolf pack range area). The SAMC framework produced spatially explicit predictions of wolf dispersal, philopatry and conflict risk ahead of recolonization prior to reintroduction efforts. Our SAMC model included: (1) movement resistance based on terrain, roads and housing density; (2) mortality risk and potential conflict (absorption) based on livestock presence, social tolerance, land ownership and state boundaries; and (3) site fidelity based on habitat quality. Using this model, we compared 21 public land units by deriving predictions of: (A) relative survival time outside each site, (B) intensity of use and retention time within each site and (C) the probability of use on adjacent public lands. We also predicted and mapped potential conflict hotspots associated with each site.Among the units assessed, a complex of USFS Wilderness areas near Aspen, chiefly the Hunter‐Fryingpan and Collegiate Peaks Wilderness areas, had the best overall rankings when comparing predictions of each metric. The area balances high‐quality, well‐connected habitat with relatively low livestock density and high social tolerance.Synthesis and applications. Our findings highlight the utility of the SAMC framework for assessing colonization areas and the capacity to identify locations for effective proactive management, especially of conflict prone species. The flexibility of the SAMC framework enables predicting likely areas of philopatry and human–wildlife conflict using spatially explicit metrics which can improve the success of conservation translocations and management of species with changing geographic extents. [ABSTRACT FROM AUTHOR]

The use of individual calving pens in modern farming is widely recognized as a good practice for promoting good animal welfare during parturition. However, determining the optimal time to move a pregnant cow to a calving pen can be a management challenge. Moving cows too early may result in prolonged occupancy of the pen, while moving them too late may increase the risk of calving complications and production-related diseases. In this paper, a simple random walk type Markov Chain Model to predict the optimal time for moving periparturient cows to individual calving pens was proposed. Behavior changes such as lying time, standing time, and rumination time were analyzed using a video monitoring system, and we formulated these changes as the states of a Markov Chain with an absorbing barrier. The model showed that the first time entering an absorbing state was the optimal time for a pregnant cow to be moved to a calving pen. The proposed method was validated through a series of experiments in a real-life dairy farm, showing promising results with high accuracy. [ABSTRACT FROM AUTHOR]

In networks, the Markov clustering (MCL) algorithm is one of the most efficient approaches in detecting clustered structures. The MCL algorithm takes as input a stochastic matrix, which depends on the adjacency matrix of the graph network under consideration. Quantum clustering algorithms are proven to be superefficient over the classical ones. Motivated by the idea of a potential clustering algorithm based on quantum Markov chains, we prove a clustering property for quantum Markov chains (QMCs) on Cayley trees associated with open quantum random walks (OQRW). [ABSTRACT FROM AUTHOR]

Quantum Markov chains (QMCs) and open quantum random walks (OQRWs) represent different quantum extensions of the classical Markov chain framework. QMCs stand as a more profound layer within the realm of Markovian dynamics. The exploration of both QMCs and OQRWs has been a predominant focus over the past decade. Recently, a significant connection between QMCs and OQRWs has been forged, yielding valuable applications. This bridge is particularly impactful when studying QMCs on tree structures, where it intersects with the realm of phase transitions in models naturally arising from quantum statistical mechanics. Furthermore, it aids in elucidating statistical properties, such as recurrence and clustering. The objective of this paper centers around delving into the intricate structure of QMCs on Cayley trees in relation to OQRWs. The foundational elements of this class of QMCs are built upon using classical probability measures that encompass the hierarchical nature of Cayley trees. This exploration unveils the pivotal role that the dynamics of OQRWs play in shaping the behavior of the Markov chains under consideration. Moreover, the analysis extends to their classical counterparts. The findings are further underscored by the examination of notable examples, contributing to a comprehensive understanding of the outcomes. [ABSTRACT FROM AUTHOR]

This exploratory study investigates a human agent's evolving judgements of reliability when interacting with an AI system. Two aims drove this investigation: (1) compare the predictive performance of quantum vs. Markov random walk models regarding human reliability judgements of an AI system and (2) identify a neural correlate of the perturbation of a human agent's judgement of the AI's reliability. As AI becomes more prevalent, it is important to understand how humans trust these technologies and how trust evolves when interacting with them. A mixed-methods experiment was developed for exploring reliability calibration in human–AI interactions. The behavioural data collected were used as a baseline to assess the predictive performance of the quantum and Markov models. We found the quantum model to better predict the evolving reliability ratings than the Markov model. This may be due to the quantum model being more amenable to represent the sometimes pronounced within-subject variability of reliability ratings. Additionally, a clear event-related potential response was found in the electroencephalographic (EEG) data, which is attributed to the expectations of reliability being perturbed. The identification of a trust-related EEG-based measure opens the door to explore how it could be used to adapt the parameters of the quantum model in real time. [ABSTRACT FROM AUTHOR]

Copyright of Journal Of Sichuan University (Natural Sciences Division) / Sichuan Daxue Xuebao-Ziran Kexueban is the property of Editorial Department of Journal of Sichuan University Natural Science Edition and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Random walks on the circle group R/Z$\mathbb {R}/\mathbb {Z}$ whose elementary steps are lattice variables with span α∉Q$\alpha \not\in \mathbb {Q}$ or p/q∈Q$p/q \in \mathbb {Q}$ taken mod Z$\mathbb {Z}$ exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup Zq$\mathbb {Z}_q$, and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper, we extend these results to random walks with irrational span α$\alpha$, and explicitly describe the transition of these Markov chains from finite to general state space as p/q→α$p/q \rightarrow \alpha$ along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a phase transition from polynomial to exponential decay after ≈q2$\approx q^2$ steps. This seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purely exponential. [ABSTRACT FROM AUTHOR]

Kemeny's constant is a meaningful index of the communicability in a network, which combines the long-term and short-term behavior of related Markov chains. Lorenzo Ciardo et al. provided an expression for Kemeny's constant of a tree obtained by concatenating some given rooted trees. In this paper, by extending some results of the expression, we provide sharp upper and lower bounds on Kemeny's constant of trees with fixed order n and matching number β. All the corresponding extremal graphs are identified. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlocal Cahn–Hilliard equation (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn–Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in L 1 of the Cahn–Hilliard Equation. We also show that the Cahn–Hilliard equation is the gradient flow of the Ginzburg–Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behavior of the solutions. [ABSTRACT FROM AUTHOR]

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $\Omega ((n/k) \log n)$ on the stationary cover time, holding for any $n$ -vertex graph $G$ and any $1 \leq k =o(n\log n)$. Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of earlier appeared work, where we have constructed Quantum Markov Chains (QMC) associated with Open Quantum Random Walks and discussed a phase transition phenomena within the proposed scheme. In the current work, we define the recurrence and accessibility of within QMC over trees. Moreover, the recurrence property of QMC associated with Open Quantum Random Walks is investigated. [ABSTRACT FROM AUTHOR]

In a socialized multi-agent system, the preferences of individuals will be inevitably influenced by others. This paper introduces an extended empathy structure to characterize the coupling process of preferences under specific relations and make it cover scenarios including human society, human–machine system, and even abiotic engineering applications. In this model, empathy is abstracted as a stochastic experience process in the form of Markov chain, and the coupled empathy utility is defined as the expectation of obtaining preferences under the corresponding probability distribution. The self-other separation is the core concept with which our structure can exhibit social attributes, including attraction of implicit states, inhibition of excessive empathy, attention of empathetic targets, and anisotropy of the utility distribution. Compared with the previous empirical models, our model has a better performance on the data set and can provide a new perspective for designing and analyzing the cognitive layer of the human–machine network, as well as the information fusion and semi-supervised clustering methods in engineering. [ABSTRACT FROM AUTHOR]

Lévy walks represent important modeling tools for a variety of real-life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far, these scaling limits have been derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here, we derive the corresponding limiting equations in the case of position-depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo–Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalizations are analyzed. [ABSTRACT FROM AUTHOR]

In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. This paper is devoted to a class of inhomogeneous random walks on $\mathbb{Z}^d$ termed 'Markov additive processes' (also known as Markov random walks, random walks with internal degrees of freedom, or semi-Markov processes). In this model, the increments of the walk are still independent but their distributions are dictated by a Markov chain, termed the internal Markov chain. While this model is largely studied in the literature, most of the results involve internal Markov chains whose operator is quasi-compact. This paper extends two results for more general internal operators: a local limit theorem and a sufficient criterion for their transience. These results are thereafter applied to a new family of models of drifted random walks on the lattice $\mathbb{Z}^d$. [ABSTRACT FROM AUTHOR]

The prediction of disaster risk paths can foresee the spread of disaster risk and lay a foundation for reducing or avoiding the harm of disaster risk. In this paper, we have improved the embedding layer of convolution neural network (CNN), and we literally refer to the specific CNN framework as IECNN. The IECNN model is proposed to predict the disaster risk path. Here, we first establish a disaster risk network in which each node represents two attributes: place and disaster type. The random walk paths are generated from the disaster risk network, and the structural characteristics of nodes in the disaster risk paths are considered in the improved embedding layer of IECNN. This study also applies Markov chain algorithm to calculate the probability of each disaster risk path and trains the IECNN model with the determined disaster risk paths. ROC curve, AUC, Accuracy, F1-measure, Precision, and Recall are used to evaluate the prediction model. In order to verify the feasibility and advantage of the proposed model, we use a dataset consisting of natural disasters in southwest China in the experimental section. Results of the comparative analysis display that the proposed model cannot only effectively predict the disaster risk path, but also provide the high performance in terms of every evaluation index. Moreover, some characters of disaster risk path are found and discussed. Resultantly, our results can provide an efficient way to prevent and control risk spread in disaster events. [ABSTRACT FROM AUTHOR]

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes. [ABSTRACT FROM AUTHOR]

Objective. A data-driven technique for parsimonious modeling and analysis of dynamic cerebral autoregulation (DCA) is developed based on the concept of diffusion maps. Specifically, first, a state-space description of DCA dynamics is considered based on arterial blood pressure, cerebral blood flow velocity, and their time derivatives. Next, an eigenvalue analysis of the Markov matrix of a random walk on a graph over the dataset domain yields a low-dimensional representation of the intrinsic dynamics. Further dimension reduction is made possible by accounting only for the two most significant eigenvalues. The value of their ratio indicates whether the underlying system is governed by active or hypoactive dynamics, indicating healthy or impaired DCA function, respectively. We assessed the reliability of the technique by considering healthy individuals and patients with unilateral internal carotid artery (ICA) stenosis or occlusion. We computed the sensitivity of the technique to detect the presumed side-to-side difference in the DCA function of the second group (assuming hypoactive dynamics on the occluded or stenotic side), using McNemar's chi square test. The results were compared with transfer function analysis (TFA). The performance of the two methods was also compared under the assumption of missing data. Main results. Both diffusion maps and TFA suggested a physiological side-to-side difference in the DCA of ICA stenosis or occlusion patients with a sensitivity of 81% and 71%, respectively. Further, both two methods suggested the difference between the occluded or stenotic side and any two sides of the healthy group. However, the diffusion maps captured additional difference between the unoccluded side and the healthy group, that TFA did not. Furthermore, compared to TFA, diffusion maps exhibited superior performance when subject to missing data. Significance. The eigenvalues ratio derived using the diffusion maps technique can be potentially used as a reliable and robust biomarker for assessing how active the intrinsic dynamics of the autoregulation is and for indicating healthy versus impaired DCA function. [ABSTRACT FROM AUTHOR]