Random Walk Dynamics Research Articles

Random Walk on T-Fractal with Stochastic Resetting

In this study, we explore the impact of stochastic resetting on the dynamics of random walks on a T-fractal network. By employing the generating function technique, we establish a recursive relation between the generating function of the first passage time (FPT) and derive the relationship between the mean first passage time (MFPT) with resetting and the generating function of the FPT without resetting. Our analysis covers various scenarios for a random walker reaching a target site from the starting position; for each case, we determine the optimal resetting probability γ* that minimizes the MFPT. We compare the results with the MFPT without resetting and find that the inclusion of resetting significantly enhances the search efficiency, particularly as the size of the network increases. Our findings highlight the potential of stochastic resetting as an effective strategy for the optimization of search processes in complex networks, offering valuable insights for applications in various fields in which efficient search strategies are crucial.

Lattice random walk dynamics with stochastic resetting in heterogeneous space

Abstract We examine the diffusive dynamics of a lattice random walk subject to resetting in a one-dimensional spatially heterogeneous environment composed of two media separated by an interface. At random times the walker may reset its position to the interface, but only when in the left medium. In addition the spatial heterogeneity results from having unequal diffusivities and biases in the two media. We construct the Master equation for the dynamics of the walker occupation probability in unbounded space, solve it exactly in terms of generating functions, and analyse the dynamics of the first and second moment. Making use of the closed form solution in the unbounded case, we build the analytic solution of the Master equation in finite and semi-infinite domains. By bounding the space on the right with a reflecting boundary we study the first-passage dynamics to a single fully absorbing target placed in the left medium away from the interface. As reset strongly increases the time to reach the target, we find that the first-passage dynamics enter the motion-limited regime even for relative small resetting probability. We also identify a surprising non-monotonic dependence of the first-passage probability mode as a function of the bias. By deriving an analytic expression for the mean first-passage time, we show when its value is independent of the diffusivity and bias in the left medium, uncovering another example of the so-called mean disorder indifference phenomenon.

Transforming data with the arcsine distribution for random walks

This article delves into a pioneering methodology for optimizing the analysis of random walk data by implementing the arcsine distribution. The application of the arcsine distribution serves as an imperceptible yet potent solution, mitigating asymmetry and introducing bounds while effectively modeling the nuanced characteristics intrinsic to random walk patterns. Through a meticulous exploration of the mathematical foundations and practical applications of this distribution, this study discreetly advances statistical methodologies for handling random walk data. The article illuminates the theoretical underpinnings, subtle advantages, and pragmatic implications of arcsine distribution utilization, showcasing its imperceptible yet impactful role in capturing and reshaping random walk dynamics. Through a meticulous exploration of the mathematical foundations and practical applications of this distribution, this study discreetly advances statistical methodologies for handling random walk data, particularly in the context of financial modeling.

Dynamic Random Walk and Dynamic Opposition Learning for Improving Aquila Optimizer: Solving Constrained Engineering Design Problems.

One of the most important tasks in handling real-world global optimization problems is to achieve a balance between exploration and exploitation in any nature-inspired optimization method. As a result, the search agents of an algorithm constantly strive to investigate the unexplored regions of a search space. Aquila Optimizer (AO) is a recent addition to the field of metaheuristics that finds the solution to an optimization problem using the hunting behavior of Aquila. However, in some cases, AO skips the true solutions and is trapped at sub-optimal solutions. These problems lead to premature convergence (stagnation), which is harmful in determining the global optima. Therefore, to solve the above-mentioned problem, the present study aims to establish comparatively better synergy between exploration and exploitation and to escape from local stagnation in AO. In this direction, firstly, the exploration ability of AO is improved by integrating Dynamic Random Walk (DRW), and, secondly, the balance between exploration and exploitation is maintained through Dynamic Oppositional Learning (DOL). Due to its dynamic search space and low complexity, the DOL-inspired DRW technique is more computationally efficient and has higher exploration potential for convergence to the best optimum. This allows the algorithm to be improved even further and prevents premature convergence. The proposed algorithm is named DAO. A well-known set of CEC2017 and CEC2019 benchmark functions as well as three engineering problems are used for the performance evaluation. The superior ability of the proposed DAO is demonstrated by the examination of the numerical data produced and its comparison with existing metaheuristic algorithms.

FlowWalker: A Memory-Efficient and High-Performance GPU-Based Dynamic Graph Random Walk Framework

Dynamic graph random walk (DGRW) emerges as a practical tool for capturing structural relations within a graph. Effectively executing DGRW on GPU presents certain challenges. First, existing sampling methods demand a pre-processing buffer, causing substantial space complexity. Moreover, the power-law distribution of graph vertex degrees introduces workload imbalance issues, rendering DGRW embarrassed to parallelize. In this paper, we propose FlowWalker, a GPU-based dynamic graph random walk framework. FlowWalker implements an efficient parallel sampling method to fully exploit the GPU parallelism and reduce space complexity. Moreover, it employs a sampler-centric paradigm alongside a dynamic scheduling strategy to handle the huge amounts of walking queries. FlowWalker stands as a memory-efficient framework that requires no auxiliary data structures in GPU global memory. We examine the performance of FlowWalker extensively on ten datasets, and experiment results show that FlowWalker achieves up to 752.2×, 72.1×, and 16.4× speedup compared with existing CPU, GPU, and FPGA random walk frameworks, respectively. Case study shows that FlowWalker diminishes random walk time from 35% to 3% in a pipeline of ByteDance friend recommendation GNN training.

Enhanced prairie dog optimization with Levy flight and dynamic opposition-based learning for global optimization and engineering design problems

This study proposes a new prairie dog optimization algorithm version called EPDO. This new version aims to address the issues of premature convergence and slow convergence that were observed in the original PDO algorithm. To improve performance, several modifications are introduced in EPDO. First, a dynamic opposite learning strategy is employed to increase the diversity of the population and prevent premature convergence. This strategy helps the algorithm avoid falling into local optima and promotes global optimization. Additionally, the Lévy dynamic random walk technique is utilized in EPDO. This modified Lévy flight with random walk reduces the algorithm’s running time for the test function’s ideal value, accelerating its convergence. The proposed approach is evaluated using 33 benchmark problems from CEC 2017 and compared against seven other comparative techniques: GWO, MFO, ALO, WOA, DA, SCA, and RSA. Numerical results demonstrate that EPDO produces good outcomes and performs well in solving benchmark problems. To further validate the results and assess reliability, the authors employ average rank tests, the measurement of alternatives, and ranking according to the compromise solution (MARCOS) method, as well as a convergence report of EPDO and other algorithms. Furthermore, the effectiveness of the EPDO algorithm is demonstrated by applying it to five design problems. The results indicate that EPDO achieves impressive outcomes and proves its capability to address practical issues. The algorithm performs well in solving benchmark and practical design problems, as supported by the numerical results and validation methods used in the study.

Chiral Skyrmions Interacting with Chiral Flowers.

The chiral nature of active matter plays an important role in the dynamics of active matter interacting with chiral structures. Skyrmions are chiral objects, and their interactions with chiral nanostructures can lead to intriguing phenomena. Here, we explore the random-walk dynamics of a thermally activated chiral skyrmion interacting with a chiral flower-like obstacle in a ferromagnetic layer, which could create topology-dependent outcomes. It is a spontaneous mesoscopic order-from-disorder phenomenon driven by the thermal fluctuations and topological nature of skyrmions that exists only in ferromagnetic and ferrimagnetic systems. The interactions between the skyrmions and chiral flowers at finite temperatures can be utilized to control the skyrmion position and distribution without applying any external driving force or temperature gradient. The phenomenon that thermally activated skyrmions are dynamically coupled to chiral flowers may provide a new way to design topological sorting devices.

Deep Direct Discriminative Decoders for High-dimensional Time-series Data Analysis

The state-space models (SSMs) are widely utilized in the analysis of time-series data. SSMs rely on an explicit definition of the state and observation processes. Characterizing these processes is not always easy and becomes a modeling challenge when the dimension of observed data grows or the observed data distribution deviates from the normal distribution. Here, we propose a new formulation of SSM for high-dimensional observation processes with a heavy-tailed distribution. We call this solution the deep direct discriminative process (D4). The D4 brings deep neural networks’ expressiveness and scalability to the SSM formulation letting us build a novel solution that efficiently estimates the underlying state processes through high-dimensional observation signal.We demonstrate the D4 solutions in simulated and real data such as Lorenz attractors, Langevin dynamics, random walk dynamics, and rat hippocampus spiking neural data and show that the D4’s performance precedes traditional SSMs and RNNs. The D4 can be applied to a broader class of time-series data where the connection between high-dimensional observation and the underlying latent process is hard to characterize.

Dynamical random walk on the integers with a drift

Dynamical random walk on the integers with a drift

Exact spatiotemporal dynamics of lattice random walks in hexagonal and honeycomb domains.

A variety of transport processes in natural and man-made systems are intrinsically random. To model their stochasticity, lattice random walks have been employed for a long time, mainly by considering Cartesian lattices. However, in many applications in bounded space the geometry of the domain may have profound effects on the dynamics and ought to be accounted for. We consider here the cases of the six-neighbor (hexagonal) and three-neighbor (honeycomb) lattices, which are utilized in models ranging from adatoms diffusing in metals and excitations diffusing on single-walled carbon nanotubes to animal foraging strategy and the formation of territories in scent-marking organisms. In these and other examples, the main theoretical tool to study the dynamics of lattice random walks in hexagonal geometries has been via simulations. Analytic representations have in most cases been inaccessible, in particular in bounded hexagons, given the complicated "zigzag" boundary conditions that a walker is subject to. Here we generalize the method of images to hexagonal geometries and obtain closed-form expressions for the occupation probability, the so-called propagator, for lattice random walks both on hexagonal and honeycomb lattices with periodic, reflective, and absorbing boundary conditions. In the periodic case, we identify two possible choices of image placement and their corresponding propagators. Using them, we construct the exact propagators for the other boundary conditions, and we derive transport-related statistical quantities such as first-passage probabilities to one or multiple targets and their means, elucidating the effect of the boundary condition on transport properties.

LightRW: FPGA Accelerated Graph Dynamic Random Walks

Graph dynamic random walks (GDRWs) have recently emerged as a powerful paradigm for graph analytics and learning applications, including graph embedding and graph neural networks. Despite the fact that many existing studies optimize the performance of GDRWs on multi-core CPUs, massive random memory accesses and costly synchronizations cause severe resource underutilization, and the processing of GDRWs is usually the key performance bottleneck in many graph applications. This paper studies an alternative architecture, FPGA, to address these issues in GDRWs, as FPGA has the ability of hardware customization so that we are able to explore fine-grained pipeline execution and specialized memory access optimizations. Specifically, we propose LightRW, a novel FPGA-based accelerator for GDRWs. LightRW embraces a series of optimizations to enable fine-grained pipeline execution on the chip and to exploit the massive parallelism of FPGA while significantly reducing memory accesses. As current commonly used sampling methods in GDRWs do not efficiently support fine-grained pipeline execution, we develop a parallelized reservoir sampling method to sample multiple vertices per cycle for efficient pipeline execution. To address the random memory access issues, we propose a degree-aware configurable caching method that buffers hot vertices on-chip to alleviate random memory accesses and a dynamic burst access engine that efficiently retrieves neighbors. Experimental results show that our optimization techniques are able to improve the performance of GDRWs on FPGA significantly. Moreover, LightRW delivers up to 9.55x and 9.10x speedup over the state-of-the-art CPU-based MetaPath and Node2vec random walks, respectively. This work is open-sourced on GitHub at https://github.com/Xtra-Computing/LightRW.

Dynamics of lattice random walk within regions composed of different media and interfaces

We study the lattice random walk dynamics in a heterogeneous space of two media separated by an interface and having different diffusivity and bias. Depending on the position of the interface, there exist two exclusive ways to model the dynamics: (a) Type A dynamics whereby the interface is placed between two lattice points, and (b) Type B dynamics whereby the interface is placed on a lattice point. For both types, we obtain exact results for the one-dimensional generating function of the Green’s function or propagator for the composite system in unbounded domain as well as domains confined with reflecting, absorbing, and mixed boundaries. For the case with reflecting confinement in the absence of bias, the steady-state probability shows a step-like behavior for the Type A dynamics, while it is uniform for the Type B dynamics. We also derive explicit expressions for the first-passage probability and the mean first-passage time, and compare the hitting time dependence to a single target. Finally, considering the continuous-space continuous-time limit of the propagator, we obtain the boundary conditions at the interface. At the interface, while the flux is the same, the probability density is discontinuous for Type A and is continuous for Type B. For the latter we derive a generalized version of the so-called leather boundary condition in the appropriate limit.

Discrete space-time resetting model: application to first-passage and transmission statistics

We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability r. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green’s function. We apply our formalism to the biased random walk dynamics in one-dimensional (1D) unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dynamics of biased random walker in 1D domain bounded with periodic and reflecting boundaries is also analyzed. Depending on the bias the first-passage probability in periodic domain shows multi-fold non-monotonicity as r is varied. Finally, we apply our formalism to study the transmission dynamics of two lattice walkers with resetting in 1D domain bounded by periodic and reflecting boundaries. The probability of a definite transmission between the walkers shows non-monotonic behavior as the resetting probabilities are varied.

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

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

Identification of Colon Cancer-Related RNAs Based on Heterogeneous Networks and Random Walk.

Simple SummaryColon cancer is a complex disease with high incidence rates and mortality worldwide. Although some medical methods have been used for screening, prevention and treatment, its molecular mechanism is still unclear. Among all dysfunctional factors, the change of mutual regulation relationship between RNAs is an important factor affecting the development of cancer. Therefore, the purpose of this study is to find RNAs related to colon cancer that have not been verified. We used differential expression analysis to screen mRNAs, miRNAs and lncRNAs and further constructed a heterogeneous interaction network among these three kinds of RNAs. The network propagation algorithm RW-DIR was then developed to mine the biological information contained in the network and to identify RNAs closely related to colon cancer. The research results have provided some theoretical support for disease research and provide a basis for narrowing the research scope of medical experiments.Colon cancer is considered as a complex disease that consists of metastatic seeding in early stages. Such disease is not simply caused by the action of a single RNA, but is associated with disorders of many kinds of RNAs and their regulation relationships. Hence, it is of great significance to study the complex regulatory roles among mRNAs, miRNAs and lncRNAs for further understanding the pathogenic mechanism of colon cancer. In this study, we constructed a heterogeneous network consisting of differentially expressed mRNAs, miRNAs and lncRNAs. This contains three kinds of vertices and six types of edges. All RNAs were re-divided into three categories, which were “related”, “irrelevant” and “unlabeled”. They were processed by dynamic excitation restart random walk (RW-DIR) for identifying colon cancer-related RNAs. Ten RNAs were finally obtained related to colon cancer, which were hsa-miR-2682-5p, hsa-miR-1277-3p, ANGPTL1, SLC22A18AS, FENDRR, PHLPP2, hsa-miR-302a-5p, APCDD1, MEX3A and hsa-miR-509-3-5p. Numerical experiments have indicated that the proposed network construction framework and the following RW-DIR algorithm are effective for identifying colon cancer-related RNAs, and this kind of analysis framework can also be easily extended to other diseases, effectively narrowing the scope of biological experimental research.

Self-Interacting Random Walks: Aging, Exploration, and First-Passage Times

Self-interacting random walks are endowed with long-range memory effects that emerge from the interaction of the random walker at time t with the territory that it has visited at earlier times t′<t. This class of non-Markovian random walks has applications in a broad range of examples, from insects to living cells, where a random walker locally modifies its environment—leaving behind footprints along its path and, in turn, responding to its own footprints. Because of their inherent non-Markovian nature, the exploration properties of self-interacting random walks have remained elusive. Here, we show that long-range memory effects can have deep consequences on the dynamics of generic self-interacting random walks; they can induce aging and nontrivial persistence and transience exponents, which we determine quantitatively, in both infinite and confined geometries. Based on this analysis, we quantify the search kinetics of self-interacting random walkers and show that the distribution of the first-passage time to a target site in a confined domain takes universal scaling forms in the large-domain size limit, which we characterize quantitatively. We argue that memory abilities induced by attractive self-interactions provide a decisive advantage for local space exploration, while repulsive self-interactions can significantly accelerate the global exploration of large domains.Received 13 July 2021Revised 8 November 2021Accepted 20 December 2021DOI:https://doi.org/10.1103/PhysRevX.12.011052Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDiffusionFluctuations & noiseRandom walksStochastic processesStatistical PhysicsInterdisciplinary Physics

Efficient coding in the economics of human brain connectomics.

In systems neuroscience, most models posit that brain regions communicate information under constraints of efficiency. Yet, evidence for efficient communication in structural brain networks characterized by hierarchical organization and highly connected hubs remains sparse. The principle of efficient coding proposes that the brain transmits maximal information in a metabolically economical or compressed form to improve future behavior. To determine how structural connectivity supports efficient coding, we develop a theory specifying minimum rates of message transmission between brain regions to achieve an expected fidelity, and we test five predictions from the theory based on random walk communication dynamics. In doing so, we introduce the metric of compression efficiency, which quantifies the trade-off between lossy compression and transmission fidelity in structural networks. In a large sample of youth (n = 1,042; age 8-23 years), we analyze structural networks derived from diffusion-weighted imaging and metabolic expenditure operationalized using cerebral blood flow. We show that structural networks strike compression efficiency trade-offs consistent with theoretical predictions. We find that compression efficiency prioritizes fidelity with development, heightens when metabolic resources and myelination guide communication, explains advantages of hierarchical organization, links higher input fidelity to disproportionate areal expansion, and shows that hubs integrate information by lossy compression. Lastly, compression efficiency is predictive of behavior-beyond the conventional network efficiency metric-for cognitive domains including executive function, memory, complex reasoning, and social cognition. Our findings elucidate how macroscale connectivity supports efficient coding and serve to foreground communication processes that utilize random walk dynamics constrained by network connectivity.

Random walks on weighted networks: a survey of local and non-local dynamics

AbstractIn this article, we present a survey of different types of random walk models with local and non-local transitions on undirected weighted networks. We present a general approach by defining the dynamics as a discrete-time Markovian process with transition probabilities expressed in terms of a symmetric matrix of weights. In the first part, we describe the matrices of weights that define local random walk dynamics like the normal random walk, biased random walks and preferential navigation, random walks in the context of digital image processing and maximum entropy random walks. In addition, we explore non-local random walks, like Lévy flights on networks, fractional transport through the new formalism of fractional graph Laplacians, and applications in the context of human mobility. Explicit relations for the stationary probability distribution, the mean first passage time and global times to characterize random walks are obtained in terms of the elements of the matrix of weights and its respective eigenvalues and eigenvectors. Finally, we apply the results to the analysis of particular local and non-local random walk dynamics, and we discuss their capacity to explore several types of networks. Our results allow us to study and compare the global dynamics of different types of random walk models.

Identifying Breast Cancer-Related Genes Based on a Novel Computational Framework Involving KEGG Pathways and PPI Network Modularity

Complex diseases, such as breast cancer, are often caused by mutations of multiple functional genes. Identifying disease-related genes is a critical and challenging task for unveiling the biological mechanisms behind these diseases. In this study, we develop a novel computational framework to analyze the network properties of the known breast cancer–associated genes, based on which we develop a random-walk-with-restart (RCRWR) algorithm to predict novel disease genes. Specifically, we first curated a set of breast cancer–associated genes from the Genome-Wide Association Studies catalog and Online Mendelian Inheritance in Man database and then studied the distribution of these genes on an integrated protein–protein interaction (PPI) network. We found that the breast cancer–associated genes are significantly closer to each other than random, which confirms the modularity property of disease genes in a PPI network as revealed by previous studies. We then retrieved PPI subnetworks spanning top breast cancer–associated KEGG pathways and found that the distribution of these genes on the subnetworks are non-random, suggesting that these KEGG pathways are activated non-uniformly. Taking advantage of the non-random distribution of breast cancer–associated genes, we developed an improved RCRWR algorithm to predict novel cancer genes, which integrates network reconstruction based on local random walk dynamics and subnetworks spanning KEGG pathways. Compared with the disease gene prediction without using the information from the KEGG pathways, this method has a better prediction performance on inferring breast cancer–associated genes, and the top predicted genes are better enriched on known breast cancer–associated gene ontologies. Finally, we performed a literature search on top predicted novel genes and found that most of them are supported by at least wet-lab experiments on cell lines. In summary, we propose a robust computational framework to prioritize novel breast cancer–associated genes, which could be used for further in vitro and in vivo experimental validation.

Diffusion geometry of multiplex and interdependent systems.

Complex networks are characterized by latent geometries induced by their topology or by the dynamics on them. In the latter case, different network-driven processes induce distinct geometric features that can be captured by adequate metrics. Random walks, a proxy for a broad spectrum of processes, from simple contagion to metastable synchronization and consensus, have been recently used, Domenico [Phys. Rev. Lett. 118, 168301 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.168301] to define the class of diffusion geometries and pinpoint the functional mesoscale organization of complex networks from a genuine geometric perspective. Here we first extend this class to families of distinct random walk dynamics-including local and nonlocal information-on multilayer networks-a paradigm for biological, neural, social, transportation, and financial systems-overcoming limitations such as the presence of isolated nodes and disconnected components, typical of real-world networks. We then characterize the multilayer diffusion geometry of synthetic and empirical systems, highlighting the role played by different random search dynamics in shaping the geometric features of the corresponding diffusion manifolds.