Probability Of Random Walk Research Articles

Detecting Fast Radio Bursts with Spectral Structure Using the Continuous Forward Algorithm

Detecting fast radio bursts (FRBs) with frequency-dependent intensity remains a challenge, as existing search algorithms do not account for the spectral shape, potentially leading to nondetections. We propose a novel detection statistic, which we call the Kalman detector, that improves the sensitivity of FRB signal detection by incorporating spectral shape information. The detection statistic is based on an optimal matched filter, marginalizing over all possible intensity functions, weighted by a random walk probability distribution, considering some decorrelation bandwidth. Our analysis of previously detected FRBs demonstrates that the Kalman score provides a comparable yet independent source of information for bursts with significant spectral structure, and the sensitivity improvement is of the order 0%–200% with a median improvement of 20%. We also applied the Kalman detector to existing data from FRB 20201124A and detected two new repeat bursts that were previously missed. Furthermore, we suggest a practical implementation for real-time surveys by employing a low significance soft-trigger from initial flux integration-based detection algorithms. The Kalman detector has the potential to significantly enhance FRB detection capabilities and enable new insights into the spectral properties of these enigmatic astrophysical phenomena.

Hyperspectral Image Classification Combining Superpixel Principal Component Analysis Dimensionality Reduction with Extended Random Walk Probability Optimization

Hyperspectral Image Classification Combining Superpixel Principal Component Analysis Dimensionality Reduction with Extended Random Walk Probability Optimization

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.

Survival probability of random walks and Lévy flights with stochastic resetting

We perform a thorough analysis of the survival probability of symmetric random walks with stochastic resetting, defined as the probability for the walker not to cross the origin up to time n. For continuous symmetric distributions of step lengths with either finite (random walks) or infinite variance (Lévy flights), this probability can be expressed in terms of the survival probability of the walk without resetting, given by Sparre Andersen theory. It is therefore universal, i.e. independent of the step length distribution. We analyze this survival probability at depth, deriving both exact results at finite times and asymptotic late-time results. We also investigate the case where the step length distribution is symmetric but not continuous, focusing our attention onto arithmetic distributions generating random walks on the lattice of integers. We investigate in detail the example of the simple Polya walk and propose an algebraic approach for lattice walks with a larger range.

Toward an Adaptive Skip-Gram Model for Network Representation Learning

The random walk process on network data is a widely-used approach for network representation learning. However, we argue that the sampling of node sequences and the subsampling for the Skip-gram’s contexts have two drawbacks. One is less possible to precisely find the most correlated context nodes for every central node with only uniform graph search. The other is not easily controlled due to the expensive cost of hyperparameter tuning. Such two drawbacks lead to higher training cost and lower accuracy due to abundant and irrelevant samples. To solve these problems, we compute the adaptive probability of random walk based on Personalized PageRank (PPR), and propose an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Adaptive SKip-gram</i> (ASK) model without using complicated sampling process and negative sampling. We utilize <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -most important neighbors for positive samples selection, and attach their corresponding PPR probability into the objective function. Based on benchmark datasets with three citation networks and three social networks, we demonstrate the improvement of our ASK model for network representation learning in tasks of link prediction, node classification, and embedding visualization. The results achieve more effective performance and efficient learning time.

Survival probability of random walks leaping over traps

We consider one-dimensional discrete-time random walks (RWs) in the presence of finite size traps of length ℓ over which the RWs can jump. We study the survival probability of such RWs when the traps are periodically distributed and separated by a distance L. We obtain exact results for the mean first-passage time and the survival probability in the special case of a double-sided exponential jump distribution. While such RWs typically survive longer than if they could not leap over traps, their survival probability still decreases exponentially with the number of steps. The decay rate of the survival probability depends in a non-trivial way on the trap length ℓ and exhibits an interesting regime when ℓ → 0 as it tends to the ratio ℓ/L, which is reminiscent of strongly chaotic deterministic systems. We generalize our model to continuous-time RWs, where we introduce a power-law distributed waiting time before each jump. In this case, we find that the survival probability decays algebraically with an exponent that is independent of the trap length. Finally, we derive the diffusive limit of our model and show that, depending on the chosen scaling, we obtain either diffusion with uniform absorption, or diffusion with periodically distributed point absorbers.

ON AN INTEGRAL OF -BESSEL FUNCTIONS AND ITS APPLICATION TO MAHLER MEASURE

AbstractCogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ , where $J_0$ is the order-zero Bessel function of the first kind and a and $r_m$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.

Delayed random walk on deterministic weighted scale-free small-world network with a deep trap

How to effectively control the trapping process in complex systems is of great importance in the study of trapping problem. Recently, the approach of delayed random walk has been introduced into several deterministic network models to steer trapping process. However, exploring delayed random walk on pseudo-fractal web with the co-evolution of topology and weight has remained out of reach. In this paper, we employ delayed random walk to guide trapping process on a salient deterministic weighted scale-free small-world network with the co-evolution of topology and weight. In greater detail, we first place a deep trap at one of initial nodes of the network. Then, a tunable parameter [Formula: see text] is introduced to modulate the transition probability of random walk and dominate the trapping process. Subsequently, trapping efficiency is used as readout of trapping process and average trapping time is employed to measure trapping efficiency. Finally, the closed form solution of average trapping time (ATT) is deduced analytically, which agrees with corresponding numerical solution. The analytical solution of ATT shows that the delayed parameter [Formula: see text] only modifies the prefactor of ATT, and keeps the leading scaling unchanged. In other words, ATT grows sublinearly with network size, whatever values [Formula: see text] takes. In summary, the work may serves as one piece of clues for modulating trapping process toward desired efficiency on more general deterministic networks.

Modeling relation paths for knowledge base completion via joint adversarial training

Knowledge Base Completion (KBC), which aims at determining the missing relations between entity pairs, has received increasing attention in recent years. Most existing KBC methods focus on either embedding the Knowledge Base (KB) into a specific semantic space or leveraging the joint probability of Random Walks (RWs) on multi-hop paths. Only a few unified models take both semantic and path-related features into consideration with adequacy. In this paper, we propose a novel method to explore the intrinsic relationship between the single relation (i.e. 1-hop path) and multi-hop paths between paired entities. We use Hierarchical Attention Networks (HANs) to select important relations in multi-hop paths and encode them into low-dimensional vectors. By treating relations and multi-hop paths as two different input sources, we use a feature extractor, which is shared by two downstream components (i.e. relation classifier and source discriminator), to capture shared/similar information between them. By joint adversarial training, we encourage our model to extract features from the multi-hop paths which are representative for relation completion. We apply the trained model (except for the source discriminator) to several large-scale KBs for relation completion. Experimental results show that our method outperforms existing path information-based approaches. Since each sub-module of our model can be well interpreted, our model can be applied to a large number of relation learning tasks.

Average trapping time on weighted directed Koch network

Numerous recent studies have focused on random walks on undirected binary scale-free networks. However, random walks with a given target node on weighted directed networks remain less understood. In this paper, we first introduce directed weighted Koch networks, in which any pair of nodes is linked by two edges with opposite directions, and weights of edges are controlled by a parameter θ . Then, to evaluate the transportation efficiency of random walk, we derive an exact solution for the average trapping time (ATT), which agrees well with the corresponding numerical solution. We show that leading behaviour of ATT is function of the weight parameter θ and that the ATT can grow sub-linearly, linearly and super-linearly with varying θ . Finally, we introduce a delay parameter p to modify the transition probability of random walk, and provide a closed-form solution for ATT, which still coincides with numerical solution. We show that in the closed-form solution, the delay parameter p can change the coefficient of ATT, but cannot change the leading behavior. We also show that desired ATT or trapping efficiency can be obtained by setting appropriate weight parameter and delay parameter simultaneously. Thereby, this work advance the understanding of random walks on directed weighted scale-free networks.

Semi-supervised graph labelling reveals increasing partisanship in the United States Congress

Graph labelling is a key activity of network science, with broad practical applications, and close relations to other network science tasks, such as community detection and clustering. While a large body of work exists on both unsupervised and supervised labelling algorithms, the class of random walk-based supervised algorithms requires further exploration, particularly given their relevance to social and political networks. This work refines and expands upon a new semi-supervised graph labelling method, the GLaSS method, that exactly calculates absorption probabilities for random walks on connected graphs. The method models graphs exactly as discrete-time Markov chains, treating labelled nodes as absorbing states. The method is applied to roll call voting data for 42 meetings of the United States House of Representatives and Senate, from 1935 to 2019. Analysis of the 84 resultant political networks demonstrates strong and consistent performance of GLaSS when estimating labels for unlabelled nodes in graphs, and reveals a significant trend of increasing partisanship within the United States Congress.

Determining average trapping time of delayed random walks on Apollonian network

Designing appropriate technology to effectively control the trapping process in complex systems and achieve the desired trapping efficiency is central in the study of trapping problem in complex systems. In this paper, we study delayed random walks on Apollonian network with a trap fixed at a given initial hub node. In more detail, a stochastic parameter p was introduced in the approach to alter the transition probability of random walks. We further derive analytically the average trapping time (ATT) as the measure of trapping efficiency with the obtained analytical expression being in good agreement with the corresponding numerical solutions. The result indicates that ATT scales sublinearly with network size when [Formula: see text]. Therefore, we introduced stochastic parameter p only alter the prefactor of ATT and left the leading scaling of ATT unchanged. Our work may pave the way for understanding how to control trapping process in real complex systems.

A serendipity-biased Deepwalk for collaborators recommendation.

Scientific collaboration has become a common behaviour in academia. Various recommendation strategies have been designed to provide relevant collaborators for the target scholars. However, scholars are no longer satisfied with the acquainted collaborator recommendations, which may narrow their horizons. Serendipity in the recommender system has attracted increasing attention from researchers in recent years. Serendipity traditionally denotes the faculty of making surprising discoveries. The unexpected and valuable scientific discoveries in science such as X-rays and penicillin may be attributed to serendipity. In this paper, we design a novel recommender system to provide serendipitous scientific collaborators, which learns the serendipity-biased vector representation of each node in the co-author network. We first introduce the definition of serendipitous collaborators from three components of serendipity: relevance, unexpectedness, and value, respectively. Then we improve the transition probability of random walk in DeepWalk, and propose a serendipity-biased DeepWalk, called Seren2vec. The walker jumps to the next neighbor node with the proportional probability of edge weight in the co-author network. Meanwhile, the edge weight is determined by the three indices in definition. Finally, Top-N serendipitous collaborators are generated based on the cosine similarity between scholar vectors. We conducted extensive experiments on the DBLP data set to validate our recommendation performance, and the evaluations from serendipity-based metrics show that Seren2vec outperforms other baseline methods without much loss of recommendation accuracy.

Effect of heterogeneous weights on the average trapping time and two types of random walks in weighted directed networks

In this paper, we define a class of weighted directed networks with the weight of its edge dominated by a weight parameter w. According to the construction of the networks, we study two types of random walks in the weighted directed networks with a trap fixed on the central node, i.e., standard random walks and mixed random walks. For the standard random walks, the trapping process is controlled by a weight parameter w which changes the transition probability of random walks. For the mixed random walks including nonnearest-neighbor hopping, the trapping process is steered by a stochastic parameter [Formula: see text], where [Formula: see text] changes the walking rule. For the above two techniques, we derive both analytically the average trapping time (ATT) as the measure of trapping efficiency, and the obtained analytical expressions are in good agreement with the corresponding numerical solutions for different values of w and [Formula: see text]. The obtained results indicate that ATT scales superlinearly with network size Nn and the weight parameter w changes simultaneously the prefactor and the leading scalings of ATT, while the stochastic parameter [Formula: see text] can only alter the prefactor of ATT and leave the leading scalings of ATT unchanged. This work may help in paving the way for understanding the effects of the link weight and nonnearest-neighbor hopping in real complex systems.

Electrical networks with prescribed current and applications to random walks on graphs

In this paper we study Current Density Impedance Imaging (CDII) on Electrical Networks. The inverse problem is to determine the conductivity matrix of an electrical network from the prescribed knowledge of the magnitude of the induced current along the edges coupled with the imposed voltage or injected current on the boundary nodes. This problem leads to a weighted $ l^1 $ minimization problem for the corresponding voltage potential. We also investigate the problem of determining the transition probabilities of random walks on graphs from the prescribed expected net number of times the walker passes along the edges of the graph. Convergent numerical algorithms for solving such problems are also presented. Our results can be utilized in the design of electrical networks when certain current flow on the network is desired as well as the design of random walk models on graphs when the expected net number of the times the walker passes along the edges is prescribed. We also show that a mass preserving flow $ J = (J_{ij}) $ on a network can be uniquely recovered from the knowledge of $ |J| = (|J_{ij}|) $ and the flux of the flow on the boundary nodes, where $ J_{ij} $ is the flow from node $ i $ to node $ j $ and $ J_{ij} = -J_{ji} $, and discuss its potential application in cryptography.

Influence of the random walk finite step on the first-passage probability

A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step length. We measure first-passage probability to touch the absorbing sphere of radius R in 2D. We found a regular deviation of the first-passage probability from the exact function, which we attribute to the finiteness of the random walk step.

Survival probability of random walks and Lévy flights on a semi-infinite line

We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, , characterized by a Lévy index , which includes standard random walks () and Lévy flights (). We study the survival probability, , representing the probability that the RW stays non-negative up to step n, starting initially at . Our main focus is on the x0-dependence of for large n. We show that displays two distinct regimes as x0 varies: (i) for (‘quantum’ regime), the discreteness of the jump process significantly alters the standard scaling behavior of and (ii) for (‘classical’ regime) the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in occurs between the quantum and the classical regime as one increases x0.

Comparing Graphs of Different Sizes

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of independent sets, among other combinatorial quantities. Our methods involve inequalities for determinants, for traces of functions of operators, and for entropy.

Asymptotic behavior of some hitting probabilities for sums of IID Gaussian random sets

ABSTRACTIn this article, we consider sequences of partial sums of iid Gaussian random sets (with respect to the Minkowski sum) and we study the asymptotic behavior of some hitting probabilities (of suitable sets of ) for these partial sums. We also illustrate the use of the importance sampling for the estimation of these hitting probabilities by Monte Carlo simulations. We obtain the analog of well-known results for level crossing probabilities of random walks, and we refer to a version of the classical Cramér's Theorem in large deviations for random compact sets existing in the literature.

Kardar–Parisi–Zhang Equation and Large Deviations for Random Walks in Weak Random Environments

We consider the transition probabilities for random walks in $1+1$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.