Random Walk Process Research Articles

Dynamic Mixture of Experts Models for Online Prediction

A mixture of experts models the conditional density of a response variable using a mixture of regression models with covariate-dependent mixture weights. We extend the finite mixture of experts model by allowing the parameters in both the mixture components and the weights to evolve in time by following random walk processes. Inference for time-varying parameters in richly parameterized mixture of experts models is challenging. We propose a sequential Monte Carlo algorithm for online inference based on a tailored proposal distribution built on ideas from linear Bayes methods and the EM algorithm. The method gives a unified treatment for mixtures with time-varying parameters, including the special case of static parameters. We assess the properties of the method on simulated data and on industrial data where the aim is to predict software faults in a continuously upgraded large-scale software project.

BDS and GPS side-lobe observation quality analysis and orbit determination with a GEO satellite onboard receiver

Multi-GNSS combination can alleviate problems associated with GNSS-based geostationary earth orbit (GEO) satellite navigation and orbit determination (OD), such as GNSS availability and observation geometry deterioration. However, only GPS has been widely applied and investigated in GEO missions, whereas GEO OD with BDS requires further exploration. The Chinese GEO satellite TJS-5, equipped with a GPS and BDS-compatible receiver, is the first GEO mission that tracks both BDS 2nd and 3rd generation satellites since BDS global deployment. With the TJS-5 real onboard data, we evaluate BDS side-lobe signal performance and the BDS contribution to GEO OD. Due to transmit antenna gain deficiencies in the side lobes, BDS shows a worse tracking performance than GPS with an average satellite number of 4.3 versus 7.8. Both GPS and BDS reveal inconsistency between carrier-phase and code observations, which reaches several meters and significantly degrades post-dynamic OD calculation. We estimate the consistency as a random walk process in the carrier-phase observation model to reduce its impact. With inconsistency estimated, the post-fit carrier-phase residuals decrease from 0.21 to 0.09 m for both GPS and BDS. The OD precision is significantly improved, from 1.95 to 1.42 m with only GPS and from 3.14 to 2.71 m with only BDS; the GPS and BDS combined OD exhibits the largest improvement from 1.74 to 0.82 m, demonstrating that adding BDS improves the OD precision by 43.3%. The above results indicate that the proposed carrier-phase inconsistency estimation approach is effective for both GPS and BDS and can achieve an orbit precision within 1.0 m using multi-GNSS measurements.

Graph Embedding Method Based on Biased Walking for Link Prediction

Link prediction is an essential and challenging problem in research on complex networks, which can provide research tools and theoretical supports for the formation and evolutionary mechanisms of networks. Existing graph representation learning methods based on random walks usually ignore the influence of local network topology on the transition probability of walking nodes when predicting the existence of links, and the sampling strategy of walking nodes during random walks is uncontrolled, which leads to the inability of these methods to effectively learn high-quality node vectors to solve the link prediction problem. To address the above challenges, we propose a novel graph embedding method for link prediction. Specifically, we analyze the evolution mechanism of links based on triadic closure theory and use the network clustering coefficient to represent the aggregation ability of the network’s local structure, and this adaptive definition of the aggregation ability of the local structure enables control of the walking strategy of nodes in the random walking process. Finally, node embedding generated based on biased walking paths is employed to solve the link prediction problem. Extensive experiments and analyses show that the TCW algorithm provides high accuracy across a diverse set of datasets.

Correlated continuous-time random walk with stochastic resetting

It is known that the introduction of stochastic resetting in an uncorrelated random walk process can lead to the emergence of a stationary state, i.e. the diffusion evolves towards a saturation state, and a steady Laplace distribution is reached. In this paper, we turn to study the anomalous diffusion of the correlated continuous-time random walk considering stochastic resetting. Results reveal that it displays quite different diffusive behaviors from the uncorrelated one. For the weak correlation case, the stochastic resetting mechanism can slow down the diffusion. However, for the strong correlation case, we find that the stochastic resetting cannot compete with the space-time correlation, and the diffusion presents the same behaviors with the one without resetting. Meanwhile, a steady distribution is never reached.

Improving Network Representation Learning via Dynamic Random Walk, Self-Attention and Vertex Attributes-Driven Laplacian Space Optimization.

Network data analysis is a crucial method for mining complicated object interactions. In recent years, random walk and neural-language-model-based network representation learning (NRL) approaches have been widely used for network data analysis. However, these NRL approaches suffer from the following deficiencies: firstly, because the random walk procedure is based on symmetric node similarity and fixed probability distribution, the sampled vertices’ sequences may lose local community structure information; secondly, because the feature extraction capacity of the shallow neural language model is limited, they can only extract the local structural features of networks; and thirdly, these approaches require specially designed mechanisms for different downstream tasks to integrate vertex attributes of various types. We conducted an in-depth investigation to address the aforementioned issues and propose a novel general NRL framework called dynamic structure and vertex attribute fusion network embedding, which firstly defines an asymmetric similarity and h-hop dynamic random walk strategy to guide the random walk process to preserve the network’s local community structure in walked vertex sequences. Next, we train a self-attention-based sequence prediction model on the walked vertex sequences to simultaneously learn the vertices’ local and global structural features. Finally, we introduce an attributes-driven Laplacian space optimization to converge the process of structural feature extraction and attribute feature extraction. The proposed approach is exhaustively evaluated by means of node visualization and classification on multiple benchmark datasets, and achieves superior results compared to baseline approaches.

On the net present value distribution affected by a random walk process

In this paper we attempt to derive properties of the distribution of the net present value NPV, when the underlying cash flow is affected by stochastic disturbances in the form of a random walk process, the disturbances thus jumping continuously between each of two values. A leverage is introduced, affecting the size of each jump in relation to time.A given time period is split into a number of sub-periods of equal length. During each sub-period, a fraction of altogether one monetary unit is transferred. This cash flow is subject to the random walk, and is discounted by a given constant discount rate. We study consequences on the distribution of the net present value as the number of sub-periods increases beyond all bounds.Two different processes are studied, which turn out to have rather different properties. In the first, Case A, the two levels between each jump, are chosen as each other's negative, and in the second, Case B, except for a timing factor, as each other's inverse. Among other consequences, it is shown that the probability distribution of the NPV in Case A is close to, but different from, a lognormal distribution, and that in Case B, the NPV indeed has a lognormal distribution, but lies stable at a constant value. Also is shown that in Case A, the leverage must be at least 1/2 for a limiting distribution to exist, and in Case B at least unity. For leverage values above these values, respectively, the random walk will not affect NPV, and the net present value becomes deterministic.

Hitting times for second-order random walks

A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is not allowed to revisit a node in one step. Second-order random walks can model physical diffusion phenomena in a more realistic way than traditional random walks and have been very successfully used in various network mining and machine learning settings. However, numerous questions are still open for this type of stochastic processes. In this work, we extend well-known results concerning mean hitting and return times of standard random walks to the second-order case. In particular, we provide simple formulas that allow us to compute these numbers by solving suitable systems of linear equations. Moreover, by introducing the ‘pullback’ first-order stochastic process of a second-order random walk, we provide second-order versions of the renowned Kac’s and Random Target Lemmas.

On the model of random walk with multiple memory structure

A model of one-dimensional random walk based on the memory flow phenomenology is constructed. In this model, the jumps of the random walk process have a convolution structure formed on the basis of a finite sequence of memory functions and a stationary, generally speaking, non-Gaussian sequence. A physical interpretation of memory functions and the stationary sequence is given. A limit theorem in the metric space D[0,1] for the normalized walk process is obtained.

Nonlinear random walks optimize the trade-off between cost and prevention in epidemics lockdown measures: The ESIR model

Contagious diseases can spread quickly in human populations, either through airborne transmission or if some other spreading vectors are abundantly accessible. They can be particularly devastating if the impact on individuals' health has severe consequences on the number of hospitalizations or even deaths. Common countermeasures to contain the epidemic spread include introducing restrictions on human interactions or their mobility in general which are often associated with an economic and social cost. In this paper, we present a targeted model of optimal social distancing on metapopulation networks, named ESIR model, which can effectively reduce the disease spreading and at the same time minimize the impact on human mobility and related costs. The proposed model is grounded in a nonlinear random walk process that considers the finite carrying capacity of the network's metanodes, the physical patches where individuals interact within mobility networks. This latter constraint is modeled as a slack compartment E for the classic SIR model and quantifies the density of vacant spaces to accommodate the diffusing individuals. Formulating the problem as a multi-objective optimization problem shows that when the walkers avoid crowded nodes, the system can rapidly approach Pareto optimality, thus reducing the spreading considerably while minimizing the impact on human mobility as also validated in empirical transport networks. These results envisage ad hoc mobility protocols that can potentially enhance policy making for pandemic control.

A real-time data assimilative forecasting system for animal tracking.

Monitoring technologies now provide real‐time animal location information, which opens up the possibility of developing forecasting systems to fuse these data with movement models to predict future trajectories. State‐space modeling approaches are well established for retrospective location estimation and behavioral inference through state and parameter estimation. Here we use a state‐space model within a comprehensive data assimilative framework for probabilistic animal movement forecasting. Real‐time location information is combined with stochastic movement model predictions to provide forecasts of future animal locations and trajectories, as well as estimates of key behavioral parameters. Implementation uses ensemble‐based sequential Monte Carlo methods (a particle filter). We first apply the framework to an idealized case using a nondimensional animal movement model based on a continuous‐time random walk process. A set of numerical forecasting experiments demonstrates the workflow and key features, such as the online estimation of behavioral parameters using state augmentation, the use of potential functions for habitat preference, and the role of observation error and sampling frequency on forecast skill. For a realistic demonstration, we adapt the framework to short‐term forecasting of the endangered southern resident killer whale (SRKW) in the Salish Sea using visual sighting information wherein the potential function reflects historical habitat utilization of SRKW. We successfully estimate whale locations up to 2.5 h in advance with a moderate prediction error (<5 km), providing reasonable lead‐in time to mitigate vessel–whale interactions. It is argued that this forecasting framework can be used to synthesize diverse data types and improve animal movement models and behavioral understanding and has the potential to lead to important advances in movement ecology.

Dynamic wake tracking based on wind turbine rotor loads and Kalman filtering

In a wind farm setting, the location of the wake to which a downwind turbine is exposed is of high interest. It can be used for closed-loop active wake control, ultimately leading to fatigue load reduction and higher power generation. This work proposes a method for dynamic tracking of the meandering wake centre. The rotor takes the role of a sensor, with its blades sampling through the incoming wind field. The measurement of the flapwise blade root bending moment and its decomposition into the non-rotating yaw and tilt moments is used. The latter are linked to the lateral and vertical wake location via a parametric model, tuned with training data from aeroelastic simulations. The implementation of an Extended Kalman Filter (EKF) adds robustness to the tracking and allows to include physical knowledge of the wake meandering to the estimation. For this, the governing equations of the dynamic wake meandering model (DWM) are used to describe the meandering motion as a random walk process. The performance, possibilities and limitations of the tracking method in various inflow conditions are shown and discussed. Generally the wake tracking works satisfying, with estimation errors below 10% of the rotor diameter under moderate turbulence intensity. The Extended Kalman Filter formulation provides the confidence in the tracked wake position. The work shows how this can be effectively used for wake impingement detection.

Exploring trapping problem on simplicial networks encoding higher-order interactions

Biological, social and man-made systems often harbor various types of higher-order interaction, whose presence may substantially impact the dynamics on these systems. As a special random walk, paradigmatic trapping problem is still scarcely explored on networks encoding higher-order interactions. In this paper, applying simplicial networks produced by edge corona product as test bed, we explore how the higher-order interaction impact trapping process. Specifically, an immobile trap is first deployed at an initial node of the networks and average trapping time (ATT) is derived analytically to measure trapping efficiency of standard weight-dependent random walk. Then, delayed random walk incorporating delay phenomenon into random walk process is introduced into the networks and closed form solution of ATT is still obtained for quantifying trapping efficiency. The obtained analytical solutions of ATT in both scenarios show that ATT grows sub-linearly with network size but its leading scaling is quantitatively dependent on the parameter specifying building block of networks. Besides, analytical expression of ATT obtained in the latter scenario shows that the parameter [Formula: see text] manipulating delayed random walk does not affect the leading scaling of ATT but can quantitatively modify its prefactor. Therefore, introduction of delay phenomenon still can modulate trapping efficiency quantitatively in the presence of higher-order interaction. This work may pave the way for modulating trapping process and related dynamical processes on general networks modeling higher-order interactions.

Infinite density and relaxation for Lévy walks in an external potential: Hermite polynomial approach.

Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(τ)=v_{0}/sqrt[τ]. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.

Identifying common driver modules by equilibrating coverage and mutual exclusivity across pan-cancer data

It is of significance to identify common driver modules from pan-cancer data to interpret heterogeneity of cancer, and the accumulated omic data have made it into reality. In this paper, the pan-cancer common driver module identification problem is formulated, which takes the frequency difference among various cancers into account. For solving this problem, a K-nearest neighbors based imputation algorithm KNNImp is firstly devised to infer the variation values for some potential significant missing genes. Secondly, a harmonic mean of coverage and mutual exclusivity based random walk algorithm HMCEwalk is proposed. It weights the integrated PPI network with the harmonic mean of gene coverage scores and mutual exclusion scores among various cancer types, and extracts modules through a random walk process. Experiments were implemented on both simulated data and real biological data. The experimental results on simulated data indicate that given two types of cancers, the HMCEwalk algorithm has a stronger tendency to identify a set of modules which not only mutate in a large proportion of samples of these cancers, but have close proportion of mutated samples for each cancer. The experimental results on biological data show that the presented imputation algorithm does play roles in regaining some important cancer related genes. In comparison with two state-of-the-art identification methods MEXCOwalk and DriveWays, the presented one exhibits competitive performance in most instances in terms of revealing the known cancer genes, producing modules having satisfied coverage and mutual exclusivity for each cancer. Many detected modules engage in the known cancer-related biological pathways. In addition, the presented method does recognize many cancer-associated genes omitted by methods MEXCOwalk and DriveWays.

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.

Enhanced Facilitated Diffusion of Membrane-Associating Proteins under Symmetric Confinement.

The facilitated surface diffusion of transiently adsorbing molecules in a planar confined microenvironment (i.e., slit-like confinement) is highly relevant to biological phenomena, such as extracellular signaling, as well as numerous biotechnology systems. Here, we studied the surface diffusion of individual proteins confined between two symmetric lipid bilayer membranes, under a continuum of confinement heights, using single-molecule tracking and convex lens-induced confinement as well as hybrid, kinetic Monte Carlo simulations of a generalized continuous time random walk process. Surface diffusion was observed to vary non-monotonically with confinement height, exhibiting a maximum at a height of ∼750 nm, where diffusion was nearly 40% greater than that for a semi-infinite system. This demonstrated that planar confinement can, in fact, increase surface diffusion, qualitatively validating previous theoretical predictions. Simulations reproduced the experimental results and suggested that confinement enhancement of surface diffusion for symmetric systems is limited to cases where the adsorbate exhibits weak surface sticking.

Average trapping time on a type of horizontally segmented three dimensional Sierpinski gasket network with two types of locally self-similar structures

As a classic self-similar network model, Sierpinski gasket network has been used many times to study the characteristics of self-similar structure and its influence on the dynamic properties of the network. However, the network models studied in these problems only contain a single self-similar structure, which is inconsistent with the structural characteristics of the actual network models. In this paper, a type of horizontally segmented three dimensional Sierpinski gasket network is constructed, whose main feature is that it contains the locally self-similar structures of the two dimensional Sierpinski gasket network and the three dimensional Sierpinski gasket network at the same time, and the scale transformation between the two kinds of self-similar structures can be controlled by adjusting the crosscutting coefficient. The analytical expression of the average trapping time of a random walker by a particular site on the network model is solved, which used to analyze the effect of two types of self-similar structures on the properties of random walks. Finally, we conclude that the dominant self-similar structure will exert a greater influence on the random walk process on the network.

A Network Sampling Strategy Inspired by Epidemic Spreading

Nowadays, network sampling has become an indispensable premise and foundation for large-scale network analysis, and its effectiveness determines to a large extent the reliability and practicability of the subsequent network analysis results. In this paper, we propose a network sampling algorithm inspired by an epidemic spreading model named the contact process. The contact process is similar to the random walk process but different from it in two key points. First, at each time step, a randomly selected sampled node rather than the latest sampled node is responsible for recruiting a new node from its neighborhood. Second, the responsible node recruits one of its neighbor nodes with a probability inversely proportional to the degree of this neighbor node, instead of equal probability. Experiments on nine indiscriminately selected real-world networks show that our proposed sampling algorithm has a significant advantage in preserving two basic network properties, the degree distributions and clustering coefficient distributions of original networks, compared with seven classical sampling methods.

Modeling time-varying natural mortality in size-structured assessment models

Temporal variation in natural mortality (M) has been recognized for having a degrading effect on the performance of stock assessment models when it is not accounted for in the model. However, time-invariant M has remained widely practiced in stock assessments due to the difficulties in estimation. Therefore, in this study we conducted simulation-estimation experiments to (1) evaluate the ability of a size-structured assessment model to reliably estimate time-varying M as well as other population quantities, and (2) quantify the consequences of mis-specifying M in a size-structured assessment model. We modified an existing size-structured assessment model to include three different approaches to estimate time-varying M within the model. A wide range of scenarios regarding the underlying M patterns (variation over time and length) was considered in this study. The performance of estimation methods largely depends upon whether the assumption of length-dependent/-independent M was correctly specified in the assessment model. Among the three methods, estimating time-varying M as a mean (representing an average process) and time-specific deviations using auxiliary information performed best. Estimating time-varying M as a random walk process also performed well except when underlying time-varying M had large year-to-year variation. A temporal pattern in M affected the ability of the assessment model to provide reliable estimates of M, spawning stock biomass and recruitment. Assuming a constant M (i.e., fixing M at the correct value) was quite robust over a range of actual temporal variation in underlying M.

An influent generator for WRRF design and operation based on a recurrent neural network with multi-objective optimization using a genetic algorithm.

Nowadays, modelling, automation and control are widely used for Water Resource Recovery Facilities (WRRF) upgrading and optimization. Influent generator (IG) models are used to provide relevant input time series for dynamic WRRF simulations used in these applications. Current IG models found in literature are calibrated on the basis of a single performance criterion, such as the mean percentage error or the root mean square error. This results in the IG being adequate on average but with a lack of representativeness of, for instance, the observed temporal variability of the dataset. However, adequately capturing influent variability may be important for certain types of WRRF optimization, e.g., reaction to peak loads, control system performance evaluation, etc. Therefore, in this study, a data-driven IG model is developed based on the long short-term memory (LSTM) recurrent neural network and is optimized by a multi-objective genetic algorithm for both mean percentage error and variability. Hence, the influent generator model is able to generate a time series with a probability distribution that better represents reality, thus giving a better influent description for WRRF design and operation. To further increase the variability of the generated time series and in this way approximate the true variability better, the model is extended with a random walk process.