Random Walk Research Articles

Improved GPS tropospheric path delay estimation using variable random walk process noise

Accurate positioning using the Global Positioning System relies on accurate modeling of tropospheric delay. Estimated tropospheric delay must vary sufficiently to capture true variations; otherwise, systematic errors propagate into estimated positions, particularly the vertical. However, if the allowed delay variation is too large, the propagation of data noise into all parameters is amplified, reducing precision. Here we investigate the optimal choice of tropospheric constraints applied in the GipsyX software, which are specified by values of random walk process noise. We use the variability of 5-min estimated positions as a proxy for tropospheric error. Given that weighted mean 5-min positions closely replicate 24-h solutions, our ultimate goal is to improve 24-h positions and other daily products, such as precise orbit parameters. The commonly adopted default constraint for the zenith wet delay (ZWD) is 3 mm/√(hr) for 5-min data intervals. Using this constraint, we observe spurious wave-like patterns of 5-min vertical displacement estimates with amplitudes ~ 100 mm coincident with Winter Storm Ezekiel of November 27, 2019, across the central/eastern USA. Loosening the constraint suppresses the spurious waves and reduces 5-min vertical displacement variability while improving water vapor estimates. Further improvement can be achieved when optimizing constraints regionally, or for each station. Globally, results are typically optimized in the range of 6–12 mm/√(hr). Generally, we at least recommend loosening the constraint from the current default of 3 mm/√(hr) to 6 mm/√(hr) for ZWD every 300 s. Constraint values must be scaled by √(x/300) for alternative data intervals of x seconds.

Relation semantic fusion in subgraph for inductive link prediction in knowledge graphs

Inductive link prediction (ILP) in knowledge graphs (KGs) aims to predict missing links between entities that were not seen during the training phase. Recent some subgraph-based methods have shown some advancements, but they all overlook the relational semantics between entities during subgraph extraction. To overcome this limitation, we introduce a novel inductive link prediction model named SASILP (Structure and Semantic Inductive Link Prediction), which comprehensively incorporates relational semantics in both subgraph extraction and node initialization processes. The model employs a random walk strategy to calculate the structural scores of neighboring nodes and utilizes an enhanced graph attention network to determine their semantic scores. By integrating both structural and semantic scores, SASILP strategically selects key nodes to form a subgraph. Furthermore, the subgraph is initialized with a node initialization technique that integrates information about neighboring relations. The experiments conducted on benchmark datasets demonstrate that SASILP outperforms state-of-the-art methods on inductive link prediction tasks, and verify the effectiveness of our approach.

Absolute Zeta functions and periodicity of quantum walks on cycles

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over $\mathbf{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.

GCFExplainer: Global Counterfactual Explainer for Graph Neural Networks

Graph neural networks (GNNs) find applications in various domains such as computational biology, natural language processing, and computer security. Owing to their popularity, there is an increasing need to explain GNN predictions since GNNs are black-box machine learning models. One way to address this issue involves using counterfactual reasoning where the objective is to alter the GNN prediction by minimal changes in the input graph. Existing methods for counterfactual explanation of GNNs are limited to instance-specific local reasoning. This approach has two major limitations of not being able to offer global recourse policies and overloading human cognitive ability with too much information. In this work, we study the global explainability of GNNs through global counterfactual reasoning. Specifically, we want to find a small set of representative counterfactual graphs that explains all input graphs. Towards this goal, we propose GCF Explainer , a novel algorithm powered by vertex-reinforced random walks on an edit map of graphs with a greedy summary . Extensive experiments on real graph datasets show that the global explanation from GCF Explainer provides important high-level insights of the model behavior and achieves a 46.9% gain in recourse coverage, a 9.5% reduction in recourse cost compared to the state-of-the-art local counterfactual explainers. We also demonstrate that GCF Explainer generates explanations that are more consistent with input dataset characteristics, and is robust under adversarial attacks. In addition, K-GCF Explainer , which incorporates a graph clustering component into GCF Explainer , is introduced as a more competitive extension for datasets with a clustering structure, leading to superior performance in three out of four datasets in the experiments and better scalability.

Advancing cancer driver gene identification through an integrative network and pathway approach

ObjectiveCancer is a complex genetic disease characterized by the accumulation of various mutations, with driver genes playing a crucial role in cancer initiation and progression. Distinguishing driver genes from passenger mutations is essential for understanding cancer biology and discovering therapeutic targets. However, the majority of existing methods ignore the mutational heterogeneity and commonalities among patients, which hinders the identification of driver genes more effectively. MethodsThis study introduces MCSdriver, a novel computational model that integrates network and pathway information to prioritize the identification of cancer driver genes. MCSdriver employs a bidirectional random walk algorithm to quantify the mutual exclusivity and functional relationships between mutated genes within patient cohorts. It calculates similarity scores based on a mutual exclusivity-weighted network and pathway coverage patterns, accounting for patient-specific heterogeneity and molecular profile similarity. ResultsThis approach enhances the accuracy and quality of driver gene identification. MCSdriver demonstrates superior performance in identifying cancer driver genes across four cancer types from The Cancer Genome Atlas, showing a higher F-score, Recall and Precision compared to existing ranking list-based and module-based models. ConclusionThe MCSdriver model not only outperforms other models in identifying known cancer driver genes but also effectively identifies novel driver genes involved in cancer-related biological processes. The model’s consideration of patient-specific heterogeneity and similarity in molecular profiles significantly enhances the accuracy and quality of driver gene identification. Validation through Gene Ontology enrichment analysis and literature mining further underscores its potential application value in personalized cancer therapy, offering a promising tool for advancing our understanding and treatment of cancer.

The detection of magnetic gradient aided by correlated random walk

We propose a novel method for sensing the gradient direction of a weak magnetic field overwhelmed by a noisy background through studying the correlated random walk of Brown particles that carry with magnetic moment. Using numerical simulation, we find that the snapshot of the distribution of Brown particles that touched the top surface of a cubic container provides useful information, i.e., the center of the distribution pattern will deflect towards the direction of magnetic gradient. We also find that the impact of magnetic noise can be effectively suppressed by the presence of correlation when the concentration of Brown particles is in the region of 0.2ρ0<ρ<0.7ρ0. Our findings may motivate novel scheme to identify the direction of magnetic gradient in the presence of strong noise, which is undoubtedly important for practical applications.

Asymptotic behavior of heat kernels and Green functions on affine buildings

We compute asymptotic formulas for the transition densities p(n;x,y) of finite range isotropic random walks on affine buildings. We also describe the asymptotic behavior of the corresponding Green functions.

GEMimp: An Accurate and Robust Imputation Method for Microbiome Data Using Graph Embedding Neural Network

Microbiome research has increasingly underscored the profound link between microbial compositions and human health, with numerous studies establishing a strong correlation between microbiome characteristics and various diseases. However, the analysis of microbiome data is frequently compromised by inherent sparsity issues, characterized by a substantial presence of observed zeros. These zeros not only skew the abundance distribution of microbial species but also undermine the reliability of scientific conclusions drawn from such data. Addressing this challenge, we introduce GEMimp, an innovative imputation method designed to infuse robustness into microbiome data analysis. GEMimp leverages the node2vec algorithm, which incorporates both Breadth-First Search (BFS) and Depth-First Search (DFS) strategies in its random walks sampling process. This approach enables GEMimp to learn nuanced, low-dimensional representations of each taxonomic unit, facilitating the reconstruction of their similarity networks with unprecedented accuracy.Our comparative analysis pits GEMimp against state-of-the-art imputation methods including SAVER, MAGIC and mbImpute. The results unequivocally demonstrate that GEMimp outperforms its counterparts by achieving the highest Pearson correlation coefficient when compared to the original raw dataset. Furthermore, GEMimp shows notable proficiency in identifying significant taxa, enhancing the detection of disease-related taxa and effectively mitigating the impact of sparsity on both simulated and real-world datasets, such as those pertaining to Type 2 Diabetes (T2D) and Colorectal Cancer (CRC). These findings collectively highlight the strong effectiveness of GEMimp, allowing for better analysis on microbial data. With alleviation of sparsity issues, it could be greatly facilitated in downstream analyses and even in the field of microbiology.

Extracting High-Fidelity Smaller Scale Subgraphs of Complex Networks by Edge-Reinforced Random Walk

Extracting High-Fidelity Smaller Scale Subgraphs of Complex Networks by Edge-Reinforced Random Walk

Predicting the Ki-67 proliferation index in cervical cancer: a preliminary comparative study of four non-Gaussian diffusion-weighted imaging models combined with histogram analysis.

The prognosis for patients with cervical cancer (CC) is strongly correlated with the Ki-67 proliferation index (PI). However, the Ki-67 PI obtained through biopsy has certain limitations. The non-Gaussian distribution diffusion model of magnetic resonance imaging (MRI) may play an important role in characterizing tissue heterogeneity. At present, there are limited data available concerning the prediction of Ki-67 PI using models based on histogram features of non-Gaussian diffusion distribution. This study aimed to determine whether preoperative histogram features from multiple non-Gaussian models of diffusion-weighted imaging can predict the Ki-67 PI in patients with CC. Our cross-sectional prospective study recruited a total of 53 patients suspected of having CC who underwent 3.0-T MRI at Sun Yat-sen Memorial Hospital of Sun Yat-sen University between January 2022 and January 2023. Fifteen b values (0-4,000 s/mm2) were used for diffusion-weighted imaging. A total of nine parameters from four non-Gaussian diffusion-weighted imaging models, including continuous-time random walk (CTRW), diffusion kurtosis imaging (DKI), fractional order calculus (FROC), and intravoxel incoherent motion (IVIM), were used. Whole-tumor volumetric histogram analysis of these parameters was then obtained. In logistic regression, significant histogram characteristics were statistically examined across two groups to build the final prediction model. To assess diagnostic parameters of the proposed model in the diagnosis of the Ki-67 PI, along with the sensitivity, specificity, and diagnostic accuracy of these various parameters from the four models, receiver operating feature analysis was applied. Among the 53 patients (55.3±9.6 years, ranging from 23 to 79 years) included in the study, 15 had a Ki-67 PI ≤50% and 38 had a Ki-67 PI >50%. Univariable analysis determined that 12 histogram features were statistically different between the two groups. In multivariable logistic regression, we ultimately selected 6 histogram features to construct the final prediction model, with CTRW_α_10th percentile [odds ratio (OR) =0.955; 95% confidence interval (CI): 0.92-0.99; P=0.019], CTRW_α_robust mean absolute deviation (OR =0.893; 95% CI: 0.81-0.99; P=0.028), and CTRW_α_uniformity (OR =0.000, 95% CI: 0.00-0.90, P=0.047) being the independent predictive variables. The area under the curve of the combined prediction model was 0.845 (95% CI: 0.74-0.95), with a sensitivity of 78.9% (95% CI: 0.63-0.90), a specificity of 86.7% (95% CI: 0.60-0.98), an accuracy of 81.1% (95% CI: 0.68-0.91), a positive predictive value of 93.8% (95% CI: 0.79-0.99), and a negative predictive value of 61.9% (95% CI: 0.38-0.82). The histogram features of multiple non-Gaussian diffusion-weighted imaging can help to predict the Ki-67 PI of CC, providing a new method for the noninvasive evaluation of critical biological features of CC.

HSMAOA: An enhanced arithmetic optimization algorithm with an adaptive hierarchical structure for its solution analysis and application in optimization problems

The Arithmetic Optimization Algorithm (AOA) has recently gained significant attention as a novel meta-heuristic algorithm. However, it faces challenges such as premature convergence and entrapment in local optima when addressing complex optimization problems. To overcome these limitations, this paper proposes an enhanced AOA, termed the Self-Adaptive Hierarchical Arithmetic Optimization Algorithm (HSMAOA). The proposed method integrates three key strategies: Firstly, a spiral-guided random walk mechanism is introduced to improve global search ability. Secondly, a novel adaptive hierarchy leader and follower mechanism is proposed, which establishes a complete multi-branch tree hierarchy with decreasing branching degrees within the population, thereby increasing information exchange among population individuals to escape local optima. Finally, a differential mutation strategy based on ranked selection is introduced to enhance candidate solution quality. HSMAOA's performance was evaluated on the CEC2022 test suite against some state-of-the-art algorithms. Results, including optimization accuracy analysis, convergence curves, and various statistical tests, demonstrate HSMAOA's superior optimization capability and robustness. In addition, tests on eight engineering structure optimization problems, including the pressure vessel design problem, the multiple disk clutch brake design problem, and the step-cone pulley problem, and so forth, further validate its effectiveness. Thus, HSMAOA shows strong competitiveness in complex optimization tasks and potential for a wide range of applications, and is an advantageous and promising alternative solution for optimization problems.

Predictive Patterns and Market Efficiency: A Deep Learning Approach to Financial Time Series Forecasting

This study explores market efficiency and behavior by integrating key theories such as the Efficient Market Hypothesis (EMH), Adaptive Market Hypothesis (AMH), Informational Efficiency and Random Walk theory. Using LSTM enhanced by optimizers like Stochastic Gradient Descent (SGD), Adam, AdaGrad, and RMSprop, we analyze market inefficiencies in the Standard and Poor’s (SPX) index over a 22-year period. Our results reveal “pockets in time” that challenge EMH predictions, particularly with the AdaGrad optimizer at a size of the hidden layer (HS) of 64. Beyond forecasting, we apply the Dominguez–Lobato (DL) and General Spectral (GS) tests as part of the Martingale Difference Hypothesis to assess statistical inefficiencies and deviations from the Random Walk model. By emphasizing “informational efficiency”, we examine how quickly new information is reflected in stock prices. We argue that market inefficiencies are transient phenomena influenced by structural shifts and information flow, challenging the notion that forecasting alone can refute EMH. Additionally, we compare LSTM with ARIMA with Exponential Smoothing, and LightGBM to highlight the strengths and limitations of these models in financial forecasting. The LSTM model excels at capturing temporal dependencies, while LightGBM demonstrates its effectiveness in detecting non-linear relationships. Our comprehensive approach offers a nuanced understanding of market dynamics and inefficiencies.

A Bi-Population Competition Adaptive Interior Search Algorithm Based on Reinforcement Learning for Flexible Job Shop Scheduling Problem

In this paper, a bi-population competition adaptive interior search algorithm (BCAISA) based on a reinforcement learning strategy is proposed for the classical flexible job shop scheduling problem (FJSP) to optimize the makespan. First, the scheduling solution is represented using a machine-job-based two-segment integer encoding method, and various heuristic rules are then applied to generate the initial population. Secondly, a bi-population mechanism is introduced to partition the population into two distinct sub-populations. These sub-populations are specifically tailored for machine assignment and operation permutation, employing different search strategies respectively, aiming to facilitate an efficient implementation of parallel search. A competition mechanism is introduced to facilitate the information exchange between the two sub-populations. Thirdly, the ISA is adapted for the discrete scheduling problem by discretizing a series of search operators, which include composition optimization, mirror search, and random walk. A Q-learning-based approach is proposed to dynamically adjust a key parameter, aiming to strike a balance between the capacity for global exploration and local exploitation. Finally, extensive experiments are conducted based on 10 well-known benchmark instances of the FJSP. The design of the experiment (DOE) method is employed to determine the algorithm’s parameters. Based on the computational results, the effectiveness of four improvement strategies is first validated. The BCAISA is then compared with fifteen published algorithms. The comparative data demonstrate that our algorithm outperforms other algorithms in 50% of benchmark instances. Additionally, according to the relative percentage deviation (RPD) from the state-of-the-art results, the BCAISA also exhibits superior performance. This highlights the effectiveness of our algorithm for solving the classical FJSP. To enhance the practical application, the scope of the ISA will be broadened in future work to more complex problems in real-world scenarios.

A Lovász-Simonovits Theorem for Hypergraphs with Application to Local Clustering

We present the first analysis of diffusion on hypergraphs based on the Lovász-Simonovits theory. We demonstrate that an averaging-based diffusion operator is the appropriate generalization of the lazy random walk diffusion on 2-graphs because the diffusion rapidly converges to its stationary state from any initial state. By proving a Lovász-Simonovits-like theorem for this diffusion, we show that the diffusion rate depends on the hypergraph's conductance. To use averaging-based diffusion for clustering, we define a generalization of personalized page rank for hypergraphs, which we call ''Averaging-based Personalized Page Rank for Hypergraphs'' (APPRH). The fact that averaging-based diffusion is linear, unlike previous hypergraph diffusions used for clustering in the literature, allows us to use the Forward Push algorithm to compute APPRH efficiently. Using this method, we obtain theoretical bounds for the conductance of our clustering that are at least a constant times better than the best-known bounds in the literature. We compare our algorithm A- HyperCut against baselines on million-scale hypergraphs and find that our method is an order of magnitude faster while being competitive regarding the conductance of the local clusters produced.

Robustness and Scalability of Incomplete Virtual Pheromone Maps for Stigmergic Collective Exploration

The Swarm Guiding and Communication System (SGCS) is a decision-making and information-sharing framework for robot swarms that only needs close-range peer-to-peer communication and no centralized control. Each robot makes decisions based on an incomplete virtual pheromone map that is updated on each interaction with another robot, imitating ant colonial behavior. Similar systems rely on continuous communication with no range limitations, environment modification, or centralized control. A computer simulation is developed to assess the effectiveness and robustness of the framework in covering an area. Consistency and the time needed for 99% coverage are compared with an unbiased random walk. The pheromone approach is shown to outperfom the unbiased one regardless of number of agents. Innate resilience to individual failures is also demonstrated.

Effects of surface roughness and Reynolds number on the solute transport through three-dimensional rough-walled rock fractures under different flow regimes

In this study, the effects of surface roughness and Reynolds number (Re) on fluid flow and solute transport are investigated based on a double rough-walled fracture model that precisely represents the natural geometries of rock fractures. The double rough-walled fracture model is composed of two three-dimensional(3D) self-affine fracture surfaces generated using the improved successive random additions (SRA). Simulation of fluid flow and solute transport through the models were conducted by directly solving the Navier-Stokes equation and advection-diffusion equation (ADE), respectively. The results indicate that as the Re increases from 0.1 to 200, the flow regime changes from linear flow to nonlinear flow accompanied with the tortuous streamlines and significant eddies. Those eddies lead to the temporary stagnant zones that delay the solute migration. The increment of Re enhances the transport heterogeneity with the transport mode changing from the diffusion-dominated to the advection-dominated behavior, which is more significant in the fracture with a larger joint roughness coefficient (JRC). All breakthrough curves (BTCs) of rough-walled fractures exhibited typical non-Fickian transport characteristics with “early arrival” and “long tailing” of BTCs. Increasing the JRC and/or Re will enhances the non-Fickian transport characteristics. The ADE model is able to accurately fit the numerical BTCs and residence time distributions (RTDs) at a low Re, but fails to capture the non-Fickian transport characteristics at a large Re. In contrast, the continuous time random walk (CTRW) model provides a better fit to the numerical simulation results over the whole range of Re. Whereas, the fitting error gradually increases with increasing Re.

Entity Linking Model Based on Cascading Attention and Dynamic Graph

The purpose of entity linking is to connect entity mentions in text to real entities in the knowledge base. Existing methods focus on using the text topic, entity type, linking order, and association between entities to obtain the target entities. Although these methods have achieved good results, they ignore the exploration of candidate entities, leading to insufficient semantic information among entities. In addition, the implicit relationship and discrimination within the candidate entities also affect the accuracy of entity linking. To address these problems, we introduce information about candidate entities from Wikipedia and construct a graph model to capture implicit dependencies between different entity decisions. Specifically, we propose a cascade attention mechanism and develop a novel local entity linkage model termed CAM-LEL. This model leverages the interaction between entity mentions and candidate entities to enhance the semantic representation of entities. Furthermore, a global entity linkage model termed DG-GEL based on a dynamic graph is established to construct an entity association graph, and a random walking algorithm and entity entropy are used to extract the implicit relationships within entities to increase the differentiation between entities. Experimental results and in-depth analyses of multiple datasets show that our model outperforms other state-of-the-art models.

A combined Markov Chain Monte Carlo and Levenberg–Marquardt inversion method for heterogeneous subsurface reservoir modeling

In this study, we systematically investigate an inverse method for heterogeneous porous media to obtain porosity and permeability fields considering only production well data. The forward modeling of the input data, namely pressure and saturation fields, is based on the motion equations of a coupled Darcy flow system involving two phases of isothermal fluid flow. We discretize these equations using a multiscale finite volume simulation technique in the spatial domain, and the backward Euler method in time domain. In the inversion procedure, we combine a global optimization method, the Markov Chain Monte Carlo (MCMC) method, with a local optimization method, the Levenberg–Marquardt (LM) method. The MCMC was implemented as the Random Walk algorithm, and to generate the samples of the porosity and permeability fields, we employed Karhunen–Loève (KL) expansion of second-order stationary fields with Gaussian covariance. The coefficients of the KL expansion are estimated by minimizing the norm of the residual dependent on these fields. At the end of the MCMC iterations, we refine the KL coefficients associated with the porosity and permeability fields using the LM method. We verify in the numerical experiments that the accuracy of the porosity and permeability fields is improved by the LM refinement step.

Brownian motion in a vector space over a local field is a scaling limit

For any natural number d, the Vladimirov–Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a d-dimensional vector space V over a local field K. Just as the Laplace operator on L2(Rd) is the infinitesimal generator of Brownian motion with state space Rd, the Vladimirov–Taibleson operator on L2(V) is the infinitesimal generator of real-time Brownian motion with state space V. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted V to be the p-adic numbers.

Adaptive location and scale estimation with kernel-weighted averages

A wide variety of location and scale estimators have been developed for light-tailed distributions. Despite indisputable importance in business, finance, cybersecurity, etc., statistical estimation and inference in the presence of heavy tails have received less attention in the literature. We adopt the Kernel-Weighted Average (KWA) approach to location and scale estimation and present a set of extensive comparisons with five prominent competitors. Unlike nonparametric kernel density estimation, the optimally tuned bandwidth for KWA estimators does not necessarily converge to zero as sample size grows. We also perform a large-scale Monte Carlo simulation to search for the optimal bandwidth that minimizes the mean squared error (MSE) of KWA location and scale estimators with simulated samples from Student’s t-distribution with degrees of freedom (df) 1 , 2 , … , 30 . We further develop an adaptive technique to estimate the df that best match the observed samples using Cramér-von Mises test of goodness-of-fit. Unlike many existing methodologies, our approach is data-driven and exhibits excellent statistical performance. To illustrate this, we apply it to three real-world financial datasets containing daily closing prices of AMC Entertainment (AMC), GameStop (GME) and Meta Platforms (META) stocks to calibrate a geometric random walk model with Student’s t log-increments.