Markov Random Walk Research Articles

Any-Horizon Uniform Random Sampling and Enumeration of Constrained Scenarios for Simulation-Based Formal Verification

Model-based approaches to the verification of non-terminating Cyber-Physical Systems (CPSs) usually rely on numerical simulation of the System Under Verification (SUV) model under input scenarios of possibly varying duration, chosen among those satisfying given constraints. Such constraints typically stem from requirements (or assumptions) on the SUV inputs and its operational environment as well as from the enforcement of additional conditions aiming at, e.g., prioritising the (often extremely long) verification activity, by, e.g., focusing on scenarios explicitly exercising selected requirements, or avoiding vacuity in their satisfaction. In this setting, the possibility to efficiently sample at random (with a known distribution, e.g., uniformly) within, or to efficiently enumerate (possibly in a uniformly random order) scenarios among those satisfying the given constraints is a key enabler for the viability of the verification process, e.g., via simulation-based statistical model checking. Unfortunately, in case of non-trivial combinations of constraints, iterative approaches like Markovian random walks in the space of sequences of inputs in general fail in extracting scenarios according to a given distribution, and can be very inefficient to produce legal scenarios of interest. We show how, given a set of constraints on the input scenarios succinctly defined by finite memory monitors, a data structure (scenario generator) can be synthesised, from which any-horizon scenarios satisfying the input constraints can be efficiently extracted by (possibly uniform) random sampling or (randomised) enumeration. Our approach enables seamless support to virtually all simulation-based approaches to CPS verification, ranging from simple random testing to statistical model checking and formal (i.e., exhaustive) verification.

Joint Effects of Fragmentation and Mercury Contamination on Marsh Periwinkle (Littoraria irrorata) Movement.

There are different ways contaminants can interact and enhance the effects of habitat fragmentation, such as modifying the movement of organisms. The present study tested the hypothesis that mercury exacerbates the effects of fragmentation by affecting the movement of the marsh periwinkle Littoraria irrorata and reducing the probability of snails crossing fragmented microlandscape experimental systems. How these changes could affect the search efficiency of organisms in the long term was assessed using hidden Markov models and random walks simulations. Bayesian nonlinear models were used to analyze the effects of fragmentation and contamination on the mean speed and mean directional change of organisms. Snail movement for control and two mercury-exposure treatments were recorded in microlandscapes with six different levels of habitat cover and three landscape replicates. The results indicated that exposed organisms had lower probabilities of crossing the landscape, reduced speed, and shifts in step length distributions. Both mercury exposure and habitat fragmentation affected the movement of the marsh periwinkle. Mercury exacerbated the effects of habitat fragmentation by affecting the cognition (e.g., route planning, orientation, and spatial learning) and movement of L. irrorata. Hence, the interaction of these stressors could further reduce the functional connectivity of landscapes and reduce the search efficiency of organisms. Environ Toxicol Chem 2022;41:1742-1753. © 2022 The Authors. Environmental Toxicology and Chemistry published by Wiley Periodicals LLC on behalf of SETAC.

Effects of temperature on the movement and feeding behaviour of the large lupine beetle, Sitona gressorius

Even though the effects of insect pests on global agricultural productivity are well recognised, little is known about movement and dispersal of many species, especially in the context of global warming. This work evaluates how temperature and light conditions affect different movement metrics and the feeding rate of the large lupine beetle, an agricultural pest responsible for widespread damage in leguminous crops. By using video recordings, the movement of 384 beetles was digitally analysed under six different temperatures and light conditions in the laboratory. Bayesian linear mixed-effect models were used to analyse the data. Furthermore, the effects of temperature on the daily diffusion coefficient of beetles were estimated by using hidden Markov models and random walk simulations. Results of this work show that temperature, light conditions, and beetles’ weight were the main factors affecting the flight probability, displacement, time being active and the speed of beetles. Significant variations were also observed in all evaluated metrics. On average, beetles exposed to light conditions and higher temperatures had higher mean speed and flight probability. However, beetles tended to stay more active at higher temperatures and less active at intermediate temperatures, around 20 °C. Therefore, both the diffusion coefficient and displacement of beetles were lower at intermediate temperatures. These results show that the movement behaviour and feeding rates of beetles can present different relationships in the function of temperature. It also shows that using a single diffusion coefficient for insects in spatially explicit models may lead to over- or underestimation of pest spread.

Multimodal Feature Fusion Based Hypergraph Learning Model.

Hypergraph learning is a new research hotspot in the machine learning field. The performance of the hypergraph learning model depends on the quality of the hypergraph structure built by different feature extraction methods as well as its incidence matrix. However, the existing models are all hypergraph structures built based on one feature extraction method, with limited feature extraction and abstract expression ability. This paper proposed a multimodal feature fusion method, which firstly built a single modal hypergraph structure based on different feature extraction methods, and then extended the hypergraph incidence matrix and weight matrix of different modals. The extended matrices fuse the multimodal abstract feature and an expanded Markov random walk range during model learning, with stronger feature expression ability. However, the extended multimodal incidence matrix has a high scale and high computational cost. Therefore, the Laplacian matrix fusion method was proposed, which performed Laplacian matrix transformation on the incidence matrix and weight matrix of every model, respectively, and then conducted a weighted superposition on these Laplacian matrices for subsequent model training. The tests on four different types of datasets indicate that the hypergraph learning model obtained after multimodal feature fusion has a better classification performance than the single modal model. After Laplace matrix fusion, the average time can be reduced by about 40% compared with the extended incidence matrix, the classification performance can be further improved, and the index F1 can be improved by 8.4%.

Nonstandard diffusion under Markovian resetting in bounded domains.

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results, we find that when the waiting time probability has first and second finite moments, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset point to the boundary, or (iii) always beneficial. Instead, when the waiting time probability has the first or the two first moments diverging we find that resetting is always beneficial. Finally, we have also found that the optimal strategy to exit the domain depends on the reset rate. For low reset rates, walkers with exponential waiting times are found to be optimal, while for high reset rate, anomalous waiting times optimize the search process.

A random walk model with a mixed memory profile: Exponential and rectangular profile

The theory of Markovian random walks is consolidated and very well understood, however the theory of non-Markovian random walks presents many challenges due to its remarkably rich phenomenology. An important open problem in this context is to study how the diffusive properties of random walk processes change when memory-induced correlations are introduced. In this work we propose a model of a random walk that evolves in time according to past memories selected from rectangular (flat) and exponentially decaying memory profiles. In this mixed memory profile model, the walker remembers either the last B steps with equal a priori probability or the steps A prior to B with exponentially decaying probability, for a total number of steps equal to A+B. The diffusive behavior of the walk is numerically examined through the Hurst exponent (H). Even in the lack of exact solutions, we are still able to show that the model can be mapped onto a RW model with rectangular memory profile.

Analytic Method for Estimating Aircraft Fix Displacement From Gyroscope’s Allan-Deviation Parameters

The noise and drift requirements for a navigation-grade gyroscope are widely known, yet there is no simple analytic model of how the noise and drift of a gyroscope influence the fix displacement error (FDE) of an inertial navigation system (INS). This work derives simple analytical expressions for the cross-track and along-track errors of an aircraft whose INS consists solely of a three-axis gyroscope system with perfect knowledge of the vertical direction. The error signal of each gyroscope is Gaussian white noise and drift modeled as a first-order Markov random walk. These expressions provide a straightforward mean of calculating the FDE of an aircraft as a function of the flight duration, velocity, noise amplitude, drift amplitude, and drift’s time constant. These expressions are validated with Monte-Carlo simulations of long flights. This model quantifies the noise-versus-drift trade-off for a gyroscope in an inertial navigation system. It can save time when estimating the noise and drift that a gyroscope must exhibit to satisfy a given position-error requirement, or vice versa. They are used in particular to confirm the values, often cited without demonstration, of the noise and drift required to meet the maximum position error of an aircraft imposed by the Federal Aviation Administration’s required navigation performance 10 specification. Finally, it demonstrates that using the minimum in the measured Allan deviation of a gyroscope as a metric of the drift is incorrect, because it fails to capture the drift’s time constant. The proper metric is the maximum in the Allan deviation.

Tree builder random walk: Recurrence, transience and ballisticity

The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of Markovian and non-Markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or transient. If s is odd, the walker is ballistic, thus transient. If s is even, the walker’s behavior can be explained from local properties of the growing tree and it can be either null recurrent or it gets trapped on some limited part of the growing tree.

On the first passage time of the parabolic boundary by the Markov random walk

In this paper, we consider a family of moments of the first passage time of a parabolic boundary by a perturbed Markov random walk described by an autoregressive process A central limit theorem is proved for a perturbed Markov random walk, and the limit behavior of this family of moments of crossing the parabolic boundary by this walk is studied.

On one family of first passage times of a Markov random walk described by an autoregressive process AR(1) for nonlinear boundaries

The consider one family of first passage times of a Markov random walk described by a first-order autoregressive process AR(1) for nonlinear boundaries. Limit theorems are proved for a Markov random walk and a family of first passage times of this walk for nonlinear boundaries.

Optimal Non-Markovian Search Strategies with n-Step Memory.

Stochastic search processes are ubiquitous in nature and are expected to become more efficient when equipped with a memory, where the searcher has been before. A natural realization of a search process with long-lasting memory is a migrating cell that is repelled from the diffusive chemotactic signal that it secretes on its way, denoted as an autochemotactic searcher. To analyze the efficiency of this class of non-Markovian search processes, we present a general formalism that allows one to compute the mean first-passage time (MFPT) for a given set of conditional transition probabilities for non-Markovian random walks on a lattice. We show that the optimal choice of the n-step transition probabilities decreases the MFPT systematically and substantially with an increasing number of steps. It turns out that the optimal search strategies can be reduced to simple cycles defined by a small parameter set and that mirror-asymmetric walks are more efficient. For the autochemotactic searcher, we show that an optimal coupling between the searcher and the chemical reduces the MFPT to 1/3 of the one for a Markovian random walk.

Mobile User Trajectory Tracking for IRS Enabled Wireless Networks

In this paper, we consider an intelligent reflecting surface (IRS) enabled mobile network, where a fixed access point (AP) communicates with a mobile user (MU) via the aid of an IRS. We assume that the MU moves from one elementary square to another following a Markov random walk within a grid, and propose a maximum a posteriori (MAP) criterion to track the movement of the MU by leveraging the line-of-sight component of the IRS-MU link. Since it is infeasible to derive an explicit expression for the average probability of estimation error (APEE) for the proposed MAP criterion, we derive a closed-form upper bound for the APEE, which is used as the cost function to optimize the phase shifts of the IRS units. Considering the unit modulus constraints incurred by the IRS units, a manifold optimization (MO) method is firstly employed to gain a favorable solution to the formulated optimization problem, followed by a low-complexity codebook based solution to circumvent the high computational cost of the MO method. Our numerical results demonstrate the superior performance of the proposed IRS phase shift designs over the benchmark method.

A Preconditioned Variant of the Refined Arnoldi Method for Computing PageRank Eigenvectors

The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking web pages, then it is widely applied as a centrality measure for networks arising in various fields such as in chemistry, bioinformatics, neuroscience and social networks. For example, it can measure the node centralities of the gene-gene annotation network to evaluate the relevance of each gene with a certain disease. The networks in some fields including bioinformatics are undirected, thus the corresponding adjacency matrices are symmetry. Mathematically, the PageRank model can be stated as finding the unit positive eigenvector corresponding to the largest eigenvalue of a transition matrix built upon the linking structure. With rapid development of science and technology, the networks in real applications become larger and larger, thus the PageRank model always desires numerical algorithms with reduced algorithmic or memory complexity. In this paper, we propose a novel preconditioning approach for solving the PageRank model. This approach transforms the original PageRank eigen-problem into a new one that is more amenable to solve. We then present a preconditioned version of the refined Arnoldi method for solving this model. We demonstrate theoretically that the preconditioned Arnoldi method has higher execution efficiency and parallelism than the refined Arnoldi method. In plenty of numerical experiments, this preconditioned method exhibits noticeably faster convergence speed over its standard counterpart, especially for difficult cases with large damping factors. Besides, this superiority maintains when this technique is applied to other variants of the refined Arnoldi method. Overall, the proposed technique can give the PageRank model a faster solving process, and this will possibly improve the efficiency of researches, engineering projects and services where this model is applied.

Diffusive transport on networks with stochastic resetting to multiple nodes.

We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.

Instance Matching in Knowledge Graphs through random walks and semantics

Instance Matching (IM) is the process of matching instances that refer to the same real-world object (e.g., the same person) across different independent Knowledge Bases (KBs). This process is considered as a key step, for instance, in the integration of different KBs. In this paper, we focus on the problem of IM across different KBs represented as Knowledge Graphs (KGs). We propose SBIGMat, a novel approach for the IM problem based on Markov random walks (RW). Our approach leverages both the local and global information mutually calculated from a pairwise similarity graph. Precisely, we first build an expanded association graph consisting of pairs of IM candidates. Then, we rank each candidate pair through the stationary distribution computed from the RW on the association graph. We propose semantic and bipartite graph-based post-processing strategies that operate on the obtained random walk ranks to optimize the final assignment of co-referents. We provide a scalable distributed implementation of our approach on top of the Spark framework and we evaluate it on benchmark datasets from the instance track of the Ontology Alignment Evaluation Initiative (OAEI). The experiments show the efficiency and scalability of SBIGMat compared to several state-of-the-art IM approaches.

Temporal oscillations in preference strength provide evidence for an open system model of constructed preference

The decision process is often conceptualized as a constructive process in which a decision maker accumulates information to form preferences about the choice options and ultimately make a response. Here we examine how these constructive processes unfold by tracking dynamic changes in preference strength. Across two experiments, we observed that mean preference strength systematically oscillated over time and found that eliciting a choice early in time strongly affected the pattern of preference oscillation later in time. Preferences following choices oscillated between being stronger than those without prior choice and being weaker than those without choice. To account for these phenomena, we develop an open system dynamic model which merges the dynamics of Markov random walk processes with those of quantum walk processes. This model incorporates two sources of uncertainty: epistemic uncertainty about what preference state a decision maker has at a particular point in time; and ontic uncertainty about what decision or judgment will be observed when a person has some preference state. Representing these two sources of uncertainty allows the model to account for the oscillations in preference as well as the effect of choice on preference formation.

Outlier detection based on weighted neighbourhood information network for mixed-valued datasets

Outlier detection is of great importance in industry as unexpected errors or faults, abnormal behaviours or phenomena, etc. can occur due to a variety of human, system, and environmental reasons. To identify and analyse these rare items, events or observations can find either anomalies or novelties and, as a result, can help avoid potential unexpected consequences or improve industrial system performance. The operating data collected from industrial systems in the Industry 4.0 era are characterized as multi-attribute (e.g., both numerical and categorical) compared to previous studies. Therefore, a new outlier detection method for mixed-valued datasets based on the weighted network model is proposed in this paper. Concretely, a weighted neighbourhood information network (WNIN) is constructed by considering the neighbourhood relations and similarities among objects to represent a dataset with mixed-valued attributes (DMA). A tailored Markov random walk method is employed to detect outlier on the predefined network model. After reaching the equilibrium, the inlier score is defined according to the out-degree of nodes in the WNIN to represent the inlier degree of objects. Experiments on two real datasets and a case study illustrate the effectiveness and adaptability of the proposed method.

SHould I Stay Or Should I Go? Zero-Size Jumps in Random Walks for Lévy Flights

Abstract We study Markovian continuos-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.

A Markovian random walk model of epidemic spreading.

We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time Markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for the reproduction numbers. Then, we assume that a walker is in one of the states: susceptible, infectious, or recovered. An infectious walker remains infectious during a certain characteristic time. If an infectious walker meets a susceptible one on the same node, there is a certain probability for the susceptible walker to get infected. By implementing this hypothesis in computer simulations, we study the space-time evolution of the emerging infection patterns. Generally, random walk approaches seem to have a large potential to study epidemic spreading and to identify the pertinent parameters in epidemic dynamics.

ОЦЕНКА ИЗМЕНЕНИЯ СТРУКТУРЫ МАТЕРИАЛА ИЗДЕЛИЯ ПРИ ЕГО МЕХАНИЧЕСКОЙ ОБРАБОТКЕ

The paper proposes a method for evaluation of the structural and mechanical characteristics of the product material during machining. A process is considered in which the indicators of interest cannot be accurately determined without destroying the part, but with a certain degree of reliability can be established by indirect parameters. There is (in particular, according to the results of destructive tests) a connection between the known indirect and unknown (hidden) sought parameters, and in the most general form this connection is of an indeterminate nature. The known parameter is assumed to be continuous, the quantitative characteristic of the considered connection is the distribution function of the probability that the hidden parameter will accept a given value with a known value of the measurable parameter. In this formulation of the problem, the structural and mechanical reorganization of the material of the part during processing can be described by a Markov random walk between absorbing levels, which corresponds to the acquisition of the required properties by the part and their irreparable distortion. For a direct assessment of the hidden characteristics of the process by its known manifestations, a hidden Markov model of the considered structural and mechanical reorganization is proposed, using a continuous phase space of states. On its basis, formulas were obtained for the probabilities of favorable and unfavorable outcomes of processing, which make it possible to introduce a control action into the process at its early stages in order to increase the proportion of the yield of suitable products.