Step Of Random Walk Research Articles

TemporalHAN: Hierarchical attention-based heterogeneous temporal network embedding

Heterogeneous temporal network embedding aims to learn each node of different types of a heterogeneous temporal network in each snapshot into a low-dimensional vector representation, which can be used for various network analysis tasks such as node classification and relationship prediction. Our work proposes a novel heterogeneous temporal graph neural network embedding framework (TemporalHAN) based on hierarchical attention using a temporal convolutional network (TCN). In particular, we introduce node-level and semantic-level attention into heterogeneous graph neural networks to identify the importance of different levels between nodes. For each snapshot, we first utilise a new random walk algorithm (NRWA) to collect strongly connected heterogeneous neighbours for each node of different types and group them by node types. In addition, the algorithm utilises a damping factor to ensure that the more recent snapshots allocate more random walk steps. We then utilise node-level and semantic-level attention to learn the importance between a node and its random walk neighbour for a specific node type and learn the importance of different node-type for this node, respectively. Finally, we adopt TCN to capture the evolution information between snapshots. Experimental results on relationship prediction and node classification reveal that the TemporalHAN is competitive against diverse state-of-the-art approaches. Our code is available at https://github.com/Legendary-L/THAN.

Efficient sampling techniques for surrogate-based optimization with thermoelastic application

In many real-life models, the objective function may be very expensive to calculate or sometimes a black-box function. Therefore, a surrogate model is required to reduce the cost of the calculations. To build a robust surrogate model, a good sampling technique is needed. This paper suggests two new sampling techniques for surrogate-based optimization, depending on four random walk steps, which in turn give a good chance for building an accurate approximation for the original function. In addition to that, one of these new sampling techniques has a new feature that was not used before. The main advantage of our two suggested sampling techniques is that there is no optimization subproblem will be solved for the next sample point, which in turn saves CPU time. Furthermore, we applied our two sampling techniques to a 1D thermal shock problem for a half-space in the context of the fractional order theory of thermoelasticity to reduce the number of function evaluations and consequently CPU time. Laplace transform techniques are used to obtain the solution of the system in the frequency domain. Applying our algorithms to the few points obtained by using Laplace’s inverse, we can reach the optimal solution for all functions filed. The predictions of our techniques are discussed and compared with the classical method. 2D and 3D graphs are plotted for our numerical results. The temperature, displacement, and stress distributions are computed and represented graphically for different times and several fractional parameters.

Paradigmatic examples for testing models of optical light polarization by spheroidal dust

We present a general framework on how the polarization of radiation due to scattering, dichroic extinction, and birefringence of aligned spheroidal dust grains can be implemented and tested in 3D Monte Carlo radiative transfer (MCRT) codes. We derive a methodology for solving the radiative transfer equation governing the changes of the Stokes parameters in dust-enshrouded objects. We utilize the Müller matrix and the extinction, scattering, linear, and circular polarization cross sections of spheroidal grains as well as electrons. An established MCRT code is used, and its capabilities are extended to include the Stokes formalism. We compute changes in the polarization state of the light by scattering, dichroic extinction, and birefringence on spheroidal grains. The dependency of the optical depth and the albedo on the polarization is treated. The implementation of scattering by spheroidal grains both for random walk steps as well as for directed scattering (peel-off) are described. The observable polarization of radiation of the objects is determined through an angle binning method for photon packages that leaves the model space as well as through an inverse ray-tracing routine for the generation of images. We present paradigmatic examples for which we derive analytical solutions of the optical light polarization by spheroidal dust particles. These tests are suited for benchmark verification of MCPOLand other such codes and allow the numerical precision reached by these codes to be quantified. We demonstrate that MCPOLis in excellent agreement (within ~0.1%) of the Stokes parameters when compared to the analytical solutions.

On free energy barriers in Gaussian priors and failure of cold start MCMC for high-dimensional unimodal distributions.

We exhibit examples of high-dimensional unimodal posterior distributions arising in nonlinear regression models with Gaussian process priors for which Markov chain Monte Carlo (MCMC) methods can take an exponential run-time to enter the regions where the bulk of the posterior measure concentrates. Our results apply to worst-case initialized ('cold start') algorithms that are local in the sense that their step sizes cannot be too large on average. The counter-examples hold for general MCMC schemes based on gradient or random walk steps, and the theory is illustrated for Metropolis-Hastings adjusted methods such as preconditioned Crank-Nicolson and Metropolis-adjusted Langevin algorithm. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media

Flow and multicomponent reactive transport in saturated/unsaturated porous media are modeled by ensembles of computational particles moving on regular lattices according to specific random walk rules. The occupation number of the lattice sites is updated with a global random walk (GRW) procedure which spreads the particles from a lattice site with computational costs comparable to those for a single random walk step in sequential procedures. To cope with the nonlinearity and the degeneracy of the Richards equation the GRW flow solver uses linearization techniques similar to the L-scheme developed in finite element/volume approaches. Numerical schemes for reactive transport, coupled with the flow solver via numerical solutions for saturation and water flux, are implemented in splitting procedures. Diffusion–advection steps are solved by GRW algorithms using either biased or unbiased random walk probabilities. Since the number of particles in GRW simulations can be as large as the number of molecules involved in chemical reactions, one avoids the cumbersome problem of rescaling particle densities to approximate concentrations. Reaction steps are therefore formulated in terms of concentrations, as in deterministic approaches. The numerical convergence of the new schemes is demonstrated by comparisons with manufactured analytical solutions. Coupled flow and reactive transport problems of contaminant biodegradation described by the Monod model are further solved and the influence of flow nonlinearity/degeneracy and of the spatial heterogeneity of the medium is investigated numerically.

Mean square winding angle of Brownian motion around an impenetrable cylinder

An exact formula is derived, as an integral, for the mean square value of the winding angle ϕ (with −∞< ϕ <∞) of Brownian motion (that is, diffusion) after time t, around an infinitely long impenetrable cylinder of radius a, having started at radius R(>a) from the axis. Strikingly, for the simpler problem with a = 0, the mean square winding angle around a straight line, is long known to be instantly infinite however far away the starting point lies. The fractally small, fast, random walk steps of mathematical Brownian motion allow unbounded windings around the zero thickness of the straight line. A remedy, if it is required, is to accord the line non-zero thickness, an impenetrable cylinder, as analysed here. The problem straightaway reduces to a 2D one of winding around a disc in a plane since the axial component of the 3D Brownian motion is independent of the others. After deriving the exact mean square winding angle, the integral is evaluated in the limit of a narrow cylinder a 2 ≪ R 2, highlighting the limits of short and long diffusion times addressed by previous approximate treatments.

Stochastic sandpile on a cycle

In the stochastic sandpile (SS) model on a graph, particles interact pairwise as follows: if two particles occupy the same vertex, they must each take an independent random walk step with some probability 0 < p < 1 of not moving. These interactions continue until each site has no more than one particle on it. We provide a formal coupling between the SS and the activated random walk models, and we use the coupling to show that for the SS with n particles on the cycle graph , the system stabilizes in O(n 3) time for all initial particle configurations, provided that p(n) tends to 1 sufficiently rapidly as n → ∞.

CBRW: a novel approach for cancelable biometric template generation based on 1-D random walk.

Cancelable Biometric is a challenging research field in which security of an original biometric image is ensured by transforming the original biometric into another irreversible domain. Several approaches have been suggested in literature for generating cancelable biometric templates. In this paper, two novel and simple cancelable biometric template generation methods based on Random Walk (CBRW) have been proposed. By employing random walk and other steps given in the proposed two algorithms viz. CBRW-BitXOR and CBRW-BitCMP, the original biometric is transformed into a cancelable template. The performance of the proposed methods is compared with other state-of-the-art methods. Experiments have been performed on eight publicly available gray and color datasets i.e. CP (ear) (gray and color), UTIRIS (iris) (gray and color), ORL (face) (gray), IIT Delhi (iris) (gray and color), and AR (face) (color). Performance of the generated templates is measured in terms of Correlation Coefficient (Cr), Root Mean Square Error (RMSE), Peak Signal to Noise Ratio (PSNR), Structural Similarity (SSIM), Mean Absolute Error (MAE), Number of Pixel Change Rate (NPCR), and Unified Average Changing Intensity (UACI). By experimental results, it has been proved that proposed methods are superior than other state-of-the-art methods in qualitative as well as quantitative analysis. Furthermore, CBRW performs better on both gray as well as color images.

Contact statistics in populations of noninteracting random walkers in two dimensions.

The interaction between individuals in biological populations, dilute components of chemical systems, or particles transported by turbulent flows depends critically on their contact statistics. This work clarifies those statistics under the simplifying assumptions that the underlying motions approximate a Brownian random walk and that the particles can be treated as noninteracting. We measure the contact-interval (also called the waiting-time or interarrival-time), contact-count, and contact-duration distributions in populations of individuals undergoing noninteracting continuous-space-time random walks on a periodic two-dimensional plane (a torus) as functions of the population number density, walker radius, and random-walk step size. The contact-interval is exponentially distributed for times longer than the ballistic mean-free-collision time but not for times shorter than that, and the contact duration distribution is strongly peaked at the ballistic-crossing time for head-on collisions when the ballistic-crossing time is short compared to the mean step duration. While successive contacts between individuals are independent, the probability of repeat contact decreases with time after a previous contact. This leads to a negative duration dependence of the waiting-time interval and overdispersion of the contact-count probability density function for all time intervals. The paper demonstrates that for populations of small particles (with a walker radius that is small compared to the mean-separation or random-walk step size), the ballistic mean-free-collision interval, the ballistic-crossing time, and the random-walk-step duration can be used to construct temporal scalings which allow for common waiting-time, contact-count, and contact-duration distributions across different populations. Semi-analytic approximations for both the waiting-time and contact-duration distributions are also presented.

Identification of cancer-related module in protein-protein interaction network based on gene prioritization.

With the rapid development of deep sequencing technologies, a large amount of high-throughput data has been available for studying the carcinogenic mechanism at the molecular level. It has been widely accepted that the development and progression of cancer are regulated by modules/pathways rather than individual genes. The investigation of identifying cancer-related active modules has received an extensive attention. In this paper, we put forward an identification method ModFinder by integrating both biological networks and gene expression profiles. More concretely, a gene scoring function is devised by using the regression model with [Formula: see text]-step random walk kernel, and the genes are ranked according to both of their active scores and degrees in the PPI network. Then a greedy algorithm NSEA is introduced to find an active module with high score and strong connectivity. Experiments were performed on both simulated data and real biological one, i.e. breast cancer and cervical cancer. Compared with the previous methods SigMod, LEAN and RegMod, ModFinder shows competitive performance. It can successfully identify a well-connected module that contains a large proportion of cancer-related genes, including some well-known oncogenes or tumor suppressors enriched in cancer-related pathways.

Fractal behaviours of networks induced on infinite tree structures by random walks

Tree graphs such as Cayley trees provide a stage to support the self-organization of fractal networks by the flow of walkers from the root vertex to the outermost shell of the tree graph. This network model is a typical example that demonstrates the ability of a random process on a network to generate fractality. However, the finite scale of the tree structure assumed in the model restricts the size of fractal networks. In this study, we removed the restriction on the size of the trees by introducing a lifetime τ (number of steps of random walks) of walkers. As a result, we successfully induced a size-independent fractal structure on a tree graph without a boundary. Our numerical results show that the mean number of offspring d b of the original tree structure determines the value of the fractal box dimension db through the relation d b — 1 = (n b — 1) -θ . The lifetime τ controls the presence or absence of small-world and scale-free properties. The ideal fractal behaviour can be maintained by selecting an appropriate value of τ. The numerical results contribute to the development of a systematic method for generating fractal small-world and scale-free networks while controlling the value of the fractal box dimension. Unlike other models that use recursive rules to generate self-similar structures, this model specifically produces small-world fractal networks with scale-free properties.

Upscaling of transport through discrete fracture networks via random walk: A comparison of models.

Many models have been created for upscaled transport modeling in discrete fracture networks (DFNs). Random walk examples of these are the Markov directed random walk (MDRW), Monte Carlo solution of the Boltzmann transport equation (BTE), and the spatial Markov model (SMM). Each model handles the correlation between the random walk steps using different techniques and has successfully reproduced the results of full-resolution transport simulations in DFNs. However, their predictive capabilities under different modeling scenarios have not been compared. We construct a set of random 2D DFNs for three different fracture transmissivity distributions to comparatively evaluate model performance. We focus specifically on random walk models to determine what aspects of the space and time step distributions (e.g., correlation and coupling) must be accounted for to get the most accurate predictions. For DFNs with low heterogeneity in fracture transmissivity, accounting for correlation generally leads to less accurate predictions of transport behavior, but as the fracture transmissivity distribution widens, preferential pathways form and correlation between modeling steps becomes important, particularly for early breakthrough predictions.

Two-type annihilating systems on the complete and star graph

Red and blue particles are placed in equal proportion throughout either the complete or star graph and iteratively sampled to take simple random walk steps. Mutual annihilation occurs when particles with different colors meet. We compare the time it takes to extinguish every particle to the analogous time in the (simple to analyze) one-type setting. Additionally, we study the effect of asymmetric particle speeds.

Improvements to the macroscopic-microscopic approach of nuclear fission

The well established macroscopic-microscopic (mac-mic) description of nuclear fission enables the prediction of fission fragment yields for a broad range of fissioning systems. In this work, we present several key enhancements to this approach. We improve upon the microscopic sector of nuclear potential energy surfaces by magnifying the resolution of the Lipkin-Nogami equations and strengthening the Strutinsky procedure, thus reducing spurious effects from the continuum. We further present a novel deterministic method for calculating fission dynamics under the assumption of strongly damped nucleonic motion. Our technique directly determines the evolution of the scissioned shape distribution according to the number of random walk steps rather than the statistical accumulation of fission events. We show that our new technique is equivalent to the Metropolis random walk pioneered over the past decade by Randrup and colleagues. It further improves upon it, as we remove the need for altering the nuclear landscape via a biased potential. With our final improvement, we calculate fission fragments mass and charge distributions using particle number projection, which affords the simultaneous calculation of both mass and charge yield distributions. Fission fragments are thus calculated from the quantum mechanical $A$-body states of the potential energy surface rather than the collective mass asymmetry variable ($\alpha_{\rm g}$) of the Finite-Range Liquid-Drop Model (FRLDM) used in past work. We highlight the success of our enhancements by predicting the odd-even staggering and the charge polarization for the neutron-induced fission of ${}^{233}$U and ${}^{235}$U.

Fast Methods for Finding Multiple Effective Influencers in Real Networks.

We present scalable first hitting time methods for finding a collection of nodes that enables the fastest time for the spread of consensus in a network. That is, given a graph G = (V, E) and a natural number k, these methods find k vertices in G that minimize the sum of hitting times (expected number of steps of random walks) from all remaining vertices. Although computationally challenging for general graphs, we exploited the characteristics of real networks and utilized Monte Carlo methods to construct fast approximation algorithms that yield near-optimal solutions.

ERRATUM : Some characterizations of Pearson's Two-Unequal Step Random Walk

Some corrections are made on our paper entitled (Some characterizations of Pearson's Two-Unequal Step Random Walk), with DOI (http://dx.doi.org/10.16929/as/2020.2263.157), published in Afrika Satistika, Vol 14 (3), pp. 2263-2273. Although they do not change the final results, they can notably affect the readers.

Commuting probabilities of infinite groups

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a sequence of almost-invariant measures on G, we show that if dc_M(G) is at least a>0 then G contains a normal subgroup G' of index f(a) and a normal subgroup H of cardinality at most g(a) such that G'/H is abelian. This generalises a result for finite groups due to P. M. Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolin, Martino and Ventura. We also describe some general conditions on M under which such theorems hold. We close with an application to 'conjugacy ratios' as introduced by Cox.

Nonempirical Free Volume Viscosity Model for Alkane Lubricants under Severe Pressures.

Viscosities η and diffusion coefficients D_{s} of linear and branched alkanes at pressure 0<P<0.7 GPa and temperature T=500-600 K are calculated from molecular dynamics simulations. Combining Stokes-Einstein, free volume, and random walk concepts results in an accurate viscosity model for the considered P and T. All model parameters (hydrodynamic radius, random walk step size, and step frequency) are extracted from equilibrium molecular dynamics via microscopic ensemble averages rendering η(P,T) aparameter-free predictor for lubrication simulations.

From Sun to Interplanetary Space: What is the Pathlength of Solar Energetic Particles?

Abstract Solar energetic particles (SEPs), accelerated during solar eruptions, propagate in turbulent solar wind before being observed with in situ instruments. In order to interpret their origin through comparison with remote sensing observations of the solar eruption, we thus must deconvolve the transport effects due to the turbulent magnetic fields from the SEP observations. Recent research suggests that the SEP propagation is guided by the turbulent meandering of the magnetic fieldlines across the mean magnetic field. However, the lengthening of the distance the SEPs travel, due to the fieldline meandering, has so far not been included in SEP event analysis. This omission can cause significant errors in estimation of the release times of SEPs at the Sun. We investigate the distance traveled by the SEPs by considering them to propagate along fieldlines that meander around closed magnetic islands that are inherent in turbulent plasma. We introduce a fieldline random walk model which takes into account the physical scales associated to the magnetic islands. Our method remedies the problem of the diffusion equation resulting in unrealistically short pathlengths, and the fractal dependence of the pathlength of random walk on the length of the random-walk step. We find that the pathlength from the Sun to 1 au can be below the nominal Parker spiral length for SEP events taking place at solar longitudes 45E to 60W, whereas the western and behind-the-limb particles can experience pathlengths longer than 2 au due to fieldline meandering.

Random walk on barely supercritical branching random walk

Let $${\mathcal {T}}$$ be a supercritical Galton–Watson tree with a bounded offspring distribution that has mean $$\mu >1$$, conditioned to survive. Let $$\varphi _{\mathcal {T}}$$ be a random embedding of $${\mathcal {T}}$$ into $${\mathbb {Z}}^d$$ according to a simple random walk step distribution. Let $${\mathcal {T}}_p$$ be percolation on $${\mathcal {T}}$$ with parameter p, and let $$p_c = \mu ^{-1}$$ be the critical percolation parameter. We consider a random walk $$(X_n)_{n \ge 1}$$ on $${\mathcal {T}}_p$$ and investigate the behavior of the embedded process $$\varphi _{{\mathcal {T}}_p}(X_n)$$ as $$n\rightarrow \infty $$ and simultaneously, $${\mathcal {T}}_p$$ becomes critical, that is, $$p=p_n\searrow p_c$$. We show that when we scale time by $$n/(p_n-p_c)^3$$ and space by $$\sqrt{(p_n-p_c)/n}$$, the process $$(\varphi _{{\mathcal {T}}_p}(X_n))_{n \ge 1}$$ converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.