Restricted Random Walk Research Articles

Precursory arch-like structures explain the clogging probability in a granular hopper flow

The clogging phenomenon finds extensive application in both industrial processes and daily life events. While this broad spectrum of application motivated extensive research to identify the general factors underlying the clogging mechanism, it results in a fragmented and system-specific understanding of the entire clogging process. Therefore, it is essential to establish a holistic understanding of all contributing factors of clogging based on the microscopic physical mechanisms. In this paper, we experimentally investigate clogging of granular materials in a two-dimensional hopper flow and present a self-consistent physical mechanism of clogging based on precursory chain structures. These chain structures follow a specific modified restricted random walk, and clogging occurs when they are mechanically stable enough to withstand the flow fluctuations. We introduce a single-particle model that can explain the arch-forming probability. Our results provide insight into the microscopic mechanism behind clogging and a broader understanding of the dynamics of dense granular flow.

A Framework for the Continuous Evaluation of 3D Motion Perception in Virtual Reality.

We present a novel framework for the detection and continuous evaluation of 3D motion perception by deploying a virtual reality environment with built-in eye tracking. We created a biologically-motivated virtual scene that involved a ball moving in a restricted Gaussian random walk against a background of 1/f noise. Sixteen visually healthy participants were asked to follow the moving ball while their eye movements were monitored binocularly using the eye tracker. We calculated the convergence positions of their gaze in 3D using their fronto-parallel coordinates and linear least-squares optimization. Subsequently, to quantify 3D pursuit performance, we employed a first-order linear kernel analysis known as the Eye Movement Correlogram technique to separately analyze the horizontal, vertical and depth components of the eye movements. Finally, we checked the robustness of our method by adding systematic and variable noise to the gaze directions and re-evaluating 3D pursuit performance. We found that the pursuit performance in the motion-through depth component was reduced significantly compared to that for fronto-parallel motion components. We found that our technique was robust in evaluating 3D motion perception, even when systematic and variable noise was added to the gaze directions. The proposed framework enables the assessment of 3D Motion perception by evaluating continuous pursuit performance through eye-tracking. Our framework paves the way for a rapid, standardized and intuitive assessment of 3D motion perception in patients with various eye disorders.

Sampling Graphlets of Multiplex Networks: A Restricted Random Walk Approach

Graphlets are induced subgraph patterns that are crucial to the understanding of the structure and function of a large network. A lot of effort has been devoted to calculating graphlet statistics where random walk-based approaches are commonly used to access restricted graphs through the available application programming interfaces (APIs). However, most of them merely consider individual networks while overlooking the strong coupling between different networks. In this article, we estimate the graphlet concentration in multiplex networks with real-world applications. An inter-layer edge connects two nodes in different layers if they actually belong to the same node. The access to a multiplex network is restrictive in the sense that the upper layer allows random walk sampling, whereas the nodes of lower layers can be accessed only through the inter-layer edges and only support random node or edge sampling. To cope with this new challenge, we define a suit of two-layer graphlets and propose novel random walk sampling algorithms to estimate the proportion of all the three-node graphlets. An analytical bound on the sampling steps is proved to guarantee the convergence of our unbiased estimator. We further generalize our algorithm to explore the tradeoff between the estimated accuracy of different graphlets when the sample budget is split into different layers. Experimental evaluation on real-world and synthetic multiplex networks demonstrates the accuracy and high efficiency of our unbiased estimators.

IMPLEMENTATION OF RESOURCE DISCOVERY MECHANISMS ON PEERSIM: ENABLING UP TO ONE MILLION NODES P2P SIMULATION

The objective of this study was to implement resource discovery mechanisms onto a Peer-to-Peer (P2P) simulator. Resource discovery mechanisms such as random walk, restricted random walk, breadth-first search, intelligent-BFS, depth-first search, adaptive probabilistic search, and blackboard resource discovery mechanism had been adopted onto PeerSim. The proposed implementation has enabled the simulation for resource discovery in P2P simulator which capable of running up to one million nodes. Brief information about each resource discovery techniques that are implemented onto PeerSim are discussed and presented.

Constructive Discrepancy Minimization by Walking on the Edges

Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer [Trans. Amer. Math. Soc., 289 (1985), pp. 679--706]: In any system of $n$ sets in a universe of size $n$, there always exists a coloring which achieves discrepancy $6\sqrt{n}$. The original proof of Spencer was existential in nature and did not give an efficient algorithm to find such a coloring. Recently, a breakthrough work of Bansal [Proceedings of FOCS, 2010, pp. 3--10] gave an efficient algorithm which finds such a coloring. His algorithm was based on an SDP relaxation of the discrepancy problem and a clever rounding procedure. In this work we give a new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk we call Edge-Walk. Our algorithm and its analysis use only basic linear algebra and is truly constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

Self avoiding walk trees and laces

This paper is divided into two sections. In the first section we describe the use depth first search trees and breadth first search trees in understanding self avoiding walks and restricted random walks. While we derive our results for Euclidean lattices the method is totally general and can be used for other lattices as well. In the second section derive an expression for the number of laces in the lace expansion of a memory four restricted random walk.

Growing self avoiding walk trees

The Growing self avoiding walk model (GSAW) was proposed to explain statistical mechanics of the growth process in polymerization. Close examination of the model reveals that it is suited only for addition polymerization where only one monomer unit is added to the growing polymer chain. In this paper we propose a model for step growth or condensation polymerisation, where all of the monomer is converted first to dimer, all dimers are converted to tetramer etc. In the calculation of probabilities of the walker taking a step in a specified direction in the GSAW model the probability is the reciprocal of the number of steps available if the walker looks one step ahead. It is seen that our model for condensation polymerization GSAW can be generalized to the walker looking a finite number of steps ahead (instead of one only) for the purpose of calculation of probabilities. This is explained in detail in the first section of the paper. In the second section of the paper we explain the use of the Depth first search (DFS) algorithm in the calculation of probabilities and moments for the generalized growing self avoiding walk model. In the third section of the paper we report the exact values of the mean square end to end distances and relevant survival probabilities for the model. When GSAW was first proposed there was a controversy over whether it belonged to the same universality class as the True self avoiding walk \(\left( \upsilon =\frac{2}{d+1}\right) \) or that of the self avoiding walk \(\left( \upsilon =\frac{3}{d+2}\right) \). At that time it was conclusively shown that the GSAW model cannot belong to the same universality class as the true self avoiding walk model. However it was never conclusively shown that \(\upsilon =\frac{3}{d+2}\) for GSAW. In this paper we propose two methods of studying this problem. One method is rigorous analysis of the DFS algorithm. We explain how this algorithm can be used to study restricted random walks with finite memory and self avoiding walks. The second method proposed by us is a detailed analysis of the generalised GSAW model proposed by us. This paper is to be viewed as an introductory paper on a new model that should be of interest to both chemists and mathematicians.

Cathode spot movement on a continuous carbon fiber reinforced Cu matrix composite in vacuum

The cathode spot movements on a carbon fiber reinforced copper matrix composite (Cf/Cu) were investigated by a scanning electron microscope and a digital high-speed video camera. The composite was prepared by infiltrating a Cu alloy doped with Ti and Cr into a porous carbon/carbon composite in vacuum. It is found that the carbon fibers have a higher ability to withstand the vacuum arc erosion than the Cu matrix. The cathode spot walks randomly on the matrix, rather than on the carbon fibers. The cathode spots are restricted on the Cu matrix in Cf/Cu. The directional movement in the high-speed images is caused by the restricted random walk of the cathode spots. A gradient distribution in the size and density of the cathode spot is present on the Cu matrix. The average arc spreading velocity is estimated to be about 0.36 m/s and the transient arc spreading velocity is in the range of 0.12–0.7 m/s.

An update algorithm for restricted random walk clustering for dynamic data sets

In this article, we present a randomized dynamic cluster algorithm for large data sets. It is based on the restricted random walk cluster algorithm by Scholl and Scholl-Paschinger that has given good results in past studies. We discuss different approaches for the clustering of dynamic data sets. In contrast to most of these methods, dynamic restricted random walk clustering is also efficient for a small percentage of changes in the data set and has the additional advantage that the updates asymptotically produce the same clusters as a reclustering with the static variant; there is thus no need for any reclustering ever. In addition, the method has a relatively low computational complexity which enables it to cluster large data sets.

Earthquake Simulation by Restricted Random Walks

This article simulates earthquake slip distributions as restricted random walks. Random walks offer several unifying insights into earthquake behaviors that physically based simulations do not. With properly tailored variables, random walks generate observed power law rates of earthquake number versus earthquake magnitude (the Gutenberg–Richter relation). Curiously, b -value, the slope of this distribution, not only fixes the ratio of small to large events but also dictates diverse earthquake scaling laws such as mean slip versus fault length and moment versus mean slip. Moreover, b -value determines the overall shape and roughness of earthquake ruptures. For example, mean random walk quakes with b = −½ have elliptical slip distributions characteristic of a uniform stress drop on a crack. Random walk earthquake simulators, tuned by comparison with field data, provide improved bases for statistical inference of earthquake behavior and hazard.

RESTRICTED RANDOM WALK NUMBERS OF THE FIRST AND SECOND KIND

In a manner analogous to the definition of Stirling numbers of the first and second kind we define a set of numbers for restricted random walks of finite memory. These two sequences of counting numbers should be useful in the exact enumeration of properties of simple and restricted random walks.

Classification by restricted random walks

We define a random walk in a data set of a metric space. In order that the random walk depends on the pattern of the data, restrictions are imposed during its generation. Since such a restricted random walk investigates only a local subset of the data, a series of random walks has to be realized for describing the entire data set. An agglomerative graph-related classification method is introduced whose hierarchy is based on these restricted random walks. It is demonstrated on various examples that this new technique is able to detect efficiently clusters of different shapes without specifying the number of groups in advance.

Flow and jam of granular particles in a two-dimensional hopper

The jamming phenomenon of granular flow of mono-disperse disks in a two-dimensional hopper is studied experimentally. When the opening of the hopper d is small, jamming occurs due to formation of an arch at the hopper opening. The jamming probability as a function of the hopper opening width is measured which show a sharp decrease at some range of d. By observing the disk configurations of the arch in the jamming events, the jamming probability can be explained quantitatively using a restricted random walker model. The average horizontal and vertical spans of the jamming arch are calculated from the restricted random walk model which compares favorably with experiments.

Spectral determinant on graphs with generalized boundary conditions

The spectral determinant of the Schr\"odinger operator ($ - \Delta + V(x) $) on a graph is computed for general boundary conditions. ($\Delta$ is the Laplacian and $V(x)$ is some potential defined on the graph). Applications to restricted random walks on graphs are discussed.

General Asymptotics of the Density of a Restricted Coalescing Random Walk System

These results explore the asymptotic behavior of the density of a system of coalescing random walks where particles begin from only a subspace of the integer lattice and are allowed to walk anywhere on the lattice. They generalize results by Bramson and Griffeath from 1980.(1) Since the probability that a given site is occupied depends on how far that site is from the originating subspace, the density of the system at a given time must be re-defined. However, the general idea is still that if the density is larger than we expect at a given time, more coalescing events will occur, and the density will correct itself over time.

Restricted random walks on a graph

The problem of a restricted random walk on graphs, which keeps track of the number of immediate reversal steps, is considered by using a transfer matrix formulation. A closed-form expression is obtained for the generating function of the number ofn-step walks withr reversal steps for walks on any graph. In the case of graphs of a uniform valence, we show that our result has a probabilistic meaning, and deduce explicit expressions for the generating function in terms of the eigenvalues of the adjacency matrix. Applications to periodic lattices and the complete graph are given.

Loop entanglement of semicrystalline polyethylene in amorphous region: Diamond lattice approach

Linear polyethylenes in the amorphous region have been simulated as restricted random walks on a diamond lattice between two absorbing planes. The links among loops were studied by treating loops as oriented curves. The local conformations of polyethylene chains i.e., trans and . gauche energy differences were considered in the simulation, thereby determining the effect of crystallization temperature on the loop entanglement. It was found that the total Gauss winding and link density of linked loops increased with the thickness of the amorphous region. This result agrees with that of the cubic lattice model. The link probability decreases very slowly with the thickness of the amorphous region. On the other hand, the results presented clearly indicate that all statistical measures of linked loops decrease with temperature. Q 1999 John Wiley & Sons, Inc. J Comput Chem 20: 348)353, 1999

Classical spin in a potential field

We consider an ensemble of restricted discrete random walks in 2+1 dimensions. The restriction on the walks is such as to given particles an intrinsic angular momentum. The walks are embedded in a field which affects the mean free path of the walks. We show that the dynamics of the walks is such that second-order effects are described by a discrete form of Schrodinger's equation for particles in a potential field. This provides a classical context of the equation which is independent of its quantum context.

A modified gambler's ruin model of polyethylene chains in the amorphous region.

Polyethylene chains in the amorphous region between two crystalline lamellae M unit apart are modeled as random walks with one-step memory on a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the statistics of the restricted random walks. They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average length of loops 2M - 0.5, the average length of ties 2/3M2 + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M3 + O(M2), the variance of the tie length 28/45M4 + O(M3), and the variance of the walk length 2M3 + O(M2).

Restricted random walks on graphs

In the first part of this contribution we outline the construction of a novel matrix associated with a graph, the entries of which give the probability of a random walk over the graphG starting at sitei to reach sitej inD ij steps. HereD ij is the distance between verticesi, j. The derived matrices, to be referred to as restricted random walk matrices and labeled as RRW matrices, are non-symmetric, for trees the entries being of the form 1/p, wherep is an integer equal to 1 or larger. In the second part of the report we consider a few invariants of the RRW matrices. We will illustrate the use of one such invariant in a regression analysis. We consider the variations of the entropies in isometric octanes with skeletal changes. The derived regression, based on a single descriptor, yields the standard error of 1.26 cal K−1 mol−1 that is the smallest yet reported in the literature.