Concept Of Random Walk Research Articles

Effective resistance, random walk and algorithms

Efficiently determining the effective resistance of large-scale electrical networks is crucial for optimizing energy distribution and assessing network robustness. In this paper, we propose a novel approach that combines random walk simulations and advanced linear algebra techniques to compute the effective resistance of complex electrical networks. Our method leverages the concept of random walks to simulate the flow of current through the network, capturing its behavior under various conditions. Subsequently, we employ graph Laplacian-based linear algebra algorithms to analyze the resulting data, enabling accurate computation of the effective resistance. In our model, we aim to deploy these combined algorithm to develop a novel method to deal with such methods.

TLFSL: link prediction in multilayer social networks using trustworthy Lévy-flight semi-local random walk

Abstract As the landscape of online social networks continues to evolve, the task of expanding connections and uncovering novel relationships presents a growing complexity. Link prediction emerges as a crucial strategy, harnessing the current network dynamics to forecast future interactions among users. While traditional single-layer network link prediction models boast a storied legacy, recent attention has shifted towards tackling analogous challenges within the realm of multilayer networks. This paradigm shift underscores the critical role of extracting topological and multimodal features to effectively evaluate link weights, thereby enriching link prediction within weighted networks. Furthermore, the establishment of trustworthy pathways between users emerges as a pivotal tactic for translating unweighted similarities into meaningful weighted metrics. Leveraging the foundational principles of local random walk techniques, this paper introduces the trustworthy Lévy-flight semi-local (TLFSL) random walk framework for link prediction in multilayer social networks. By seamlessly integrating intralayer and interlayer information, TLFSL harnesses a dependable Lévy-flight random walk mechanism to anticipate new links within target layers of multilayer networks. Traditional local random walk techniques often overlook global relationships, as they confine path exploration to immediate neighbours. However, the absence of a direct edge between nodes does not necessarily imply a lack of relationship; nodes with semantic affinity may be spatially distant within the network. To overcome this limitation, we introduce the concept of semi-local random walk, which enables walker hopping with a wider global perspective. Meanwhile, TLFSL includes a distributed local community detection strategy to improve the performance of TLFSL in dealing with large-scale networks. Rigorous experimentation across diverse real-world multilayer networks consistently demonstrates TLFSL’s superior performance compared to equivalent methods.

Random walk theory and application

This project presents an overview of Random Walk Theory and its applications, as discussed in the provided project work. Random Walk Theory posits that changes in elements like stock prices follow a distribution independent of past movements, making future predictions challenging. Originating from the work of French mathematician Louis Bachelier and later popularized by economist Burton Markiel, the theory finds extensive applications beyond finance, spanning fields such as psychology, economics, and physics. The project delves into various types of random walks, including symmetric random walks, and explores their implications in different spaces, from graphs to higher-dimensional vector spaces. It provides definitions, examples, and graphical representations to elucidate random walk concepts, highlighting their relevance in practical scenarios like particle movement and stock price fluctuations. Key concepts such as the reflection principle and the main lemma are discussed to provide a comprehensive understanding of random walks and their properties. Through examples and lemmas, the project elucidates the mathematical foundations of random walks, offering insights into their behavior and applications across diverse disciplines. In summary, this project contributes to a deeper comprehension of Random Walk Theory, serving as a fundamental framework for understanding stochastic processes and their real-world implications.

Enumeration of Self-Avoiding Random Walks on Lattices as Model Chains in Polymer Crystals

Recent simulation studies have revealed a wealth of distinct crystal polymorphs encountered in the self-organization of polymer systems driven by entropy or free energy. The present analysis, based on the concept of self-avoiding random walks (SAWs) on crystal lattices, is useful to calculate upper bounds for the entropy difference of the crystals that are formed during polymer crystallization and thus to predict the thermodynamic stability of distinct polymorphs. Here, we compare two pairs of crystals sharing the same coordination number, ncoord: hexagonal close-packed (HCP) and face centered cubic (FCC), both having ncoord = 12 and the same packing density, and the less dense simple hexagonal (HEX) and body centered cubic (BCC) lattices, with ncoord = 8. In both cases, once a critical number of steps is reached, one of the crystals shows a higher number of SAWs compatible with its geometry. We explain the observed trends in terms of the bending and torsion angles as imposed by the geometric constraints of the crystal lattice.

Application of Random Walk for Disease Prediction

Disease is a great challenge to human civilization. In the long history, human beings have been fighting against diseases. In addition to finding drugs to treat diseases, predicting diseases is also the focus of human research. However, many research methods now ignore the randomness of disease. As people realize the randomness of disease, random walk technology is gradually applied to predict disease-related problems. This paper studies the characteristics of applying random walk method to predict disease-related problems. This article first introduces the concept of random walk, and then discusses the characteristics of random walk applied to problems related to genetic diseases, random walk applied to problems related to cancer, and random walk applied to problems related to the relationship between microorganisms and diseases. Specifically, it includes lncrnas, pathogenic genes, cancer genes, cancer classification, microorganisms, and diseases. The research of this article hopes to make human beings more aware of the importance of the randomness of diseases in predicting diseases. It is hoped that the characteristics of the method of applying random walk presented in this study can inspire people's future research.

Random walks through poetry

ABSTRACT The contact zone between algorithms and creative writing has become an increasingly fertile ground for literary experimentation in the last decades, with more accessible digital technologies broadening the scope of potential artistic productions and allowing the exploration of new forms of creativity. Inspired by the playful collaboration between arts and sciences embodied by the experimental creations of the French literary group OuLiPo, we borrow from physics and mathematics the concept of self-avoiding random walk to create original poems. This model is adapted here to generate random poems through potentially infinite combinations of a finite set of verses distributed on a virtual two-dimensional grid. The highly versatile algorithm we designed allows to randomly generate unexpected poems and provide, for each, a unique path with the graphical representation of a random walk.

Future assessment of the impact of the COVID-19 pandemic on the electricity market based on a stochastic socioeconomic model

This paper proposes a time-series stochastic socioeconomic model for analyzing the impact of the pandemic on the regulated distribution electricity market. The proposed methodology combines the optimized tariff model (socioeconomic market model) and the random walk concept (risk assessment technique) to ensure robustness/accuracy. The model enables both a past and future analysis of the impact of the pandemic, which is essential to prepare regulatory agencies beforehand and allow enough time for the development of efficient public policies. By applying it to six Brazilian concession areas, results demonstrate that consumers have been/will be heavily affected in general, mainly due to the high electricity tariffs that took place with the pandemic, overcoming the natural trend of the market. In contrast, the model demonstrates that the pandemic did not/will not significantly harm power distribution companies in general, mainly due to the loan granted by the regulator agency, named COVID-account. Socioeconomic welfare losses averaging 500 (MR$/month) are estimated for the equivalent concession area, i.e., the sum of the six analyzed concession areas. Furthermore, this paper proposes a stochastic optimization problem to mitigate the impact of the pandemic on the electricity market over time, considering the interests of consumers, power distribution companies, and the government. Results demonstrate that it is successful as the tariffs provided by the algorithm compensate for the reduction in demand while increasing the socioeconomic welfare of the market.

Pricing Exotic Options Using Some Lattice Procedures

In this work, we discuss some very simple and extremely efficient lattice models, namely, Binomial tree model (BTM) and Trinomial tree model (TTM) for valuing some types of exotic barrier options in details. For both these models, we consider the concept of random walks in the simulation of the path which is followed by the underlying stock price. Our main objective is to estimate the value of barrier options by using BTM and TTM for different time steps and compare these with the exact values obtained by the benchmark Black-Scholes model (BSM). Moreover, we analyze the convergence of these lattice models for these exotic options. All the results have been shown numerically as well as graphically. GANITJ. Bangladesh Math. Soc.41.1 (2021) 26-40

Clustering of graphs using pseudo-guided random walk

Clustering is an unsupervised learning task that models data as coherent groups. Multiple approaches have been proposed in the past to cluster large volumes of data. Graphs provide a logical mapping of many real-world datasets rich enough to reflect various peculiarities of numerous domains. Apart from k-means, k-medoid, and other well-known clustering algorithms, utilization of random walk-based approaches to cluster data is a prominent area of data mining research. Markov clustering algorithm and limited random walk-based clustering are the prominent techniques that utilize the concept of random walk. The main goal of this work is to address the task of clustering graphs using an efficient random walk-based method. A novel walk approach in a graph is presented here that determines the weight of the edges and the degree of the nodes. This information is utilized by the pseudo-guidance model to guide the random walk procedure. This work introduces the friends-of-friends concept during the random walk process so that the edges’ weights are determined utilizing an inclusive criterion. This concept enables a random walk to be initiated from the highest degree node. The random walk continues until the walking agent cannot find any unvisited neighbor(s). The agent walks to its neighbors if it finds a weight of one or more, otherwise the agent’s stopping criteria is met. The nodes visited in this walk form a cluster. Once a walk comes to halt, the visited nodes are removed from the original graph and the next walk starts in the remaining graph. This process continues until all nodes of the graph are traversed. The focus of this work remains random walk-based clustering of graphs. The proposed approach is evaluated using 18 real-world benchmark datasets utilizing six cluster validity indices, namely Davies-Bouldin index (DBI), Dunn index (DI), Silhouette coefficient (SC), Calinski-Harabasz index (CHI), modularity index, and normalized cut. This proposal is compared with seven closely related approaches from the same domain, namely, limited random walk, pairwise clustering, personalized page rank clustering, GAKH (genetic algorithm krill herd) graph clustering, mixing time of random walks, density-based clustering of large probabilistic graphs, and Walktrap. Experiments suggest better performance of this work based on the evaluation metrics.

Biased Random Walk in Crowded Environment: Breaking Uphill/Downhill Symmetry of Transition Times.

Various natural processes can be analyzed using the concept of random walks. For a single random walker, the mean waiting times for uphill and downhill transitions between neighboring sites are equal. Here we investigate the uphill/downhill symmetry of waiting times for transitions of a tracer in crowded environment using exactly solvable one-dimensional stochastic models. It is found that, unexpectedly, the time to move in the direction of the bias (downhill) is always longer than the time to move against the bias (uphill). The degree of asymmetry depends on the particle density, the strength of the bias, and the size of the system. The microscopic origin of the symmetry breaking is discussed.

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

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

Research on recognition and protection of ecological security patterns based on circuit theory: a case study of Jinan City.

Accelerated urbanization and population growth have resulted in the loss of ecological land and biodiversity, accompanied by the degradation of ecosystem services. Identifying and improving existing ecological security patterns are of great significance for maintaining the sustainable development of cities. In this study, Jinan, the capital of China's Shandong Province, was used as a case study area. Based on three ecosystem services, namely, soil conservation, water conservation and carbon fixation, ecological sources were determined. Furthermore, a resistance surface was constructed based on biodiversity. On these bases, the circuit theory concept of random walks was applied to simulate ecosystem processes in a heterogeneous landscape and identify ecological corridors, pinch points and barriers. A total of 25 ecological sources, 48 ecological corridors and 19 pinch points were identified, and restoration areas were delimited to three levels. These elements together constituted the ecological security patterns. Specifically, the ecological sources were mainly distributed in southern Jinan and were covered mostly with forest land. The ecological corridors were located mainly in the eastern and southwestern plains below the southern mountainous areas and were covered mostly with cropland. Furthermore, the eastern corridors were much longer than the southwestern corridors. Pinch points were distributed mostly along rivers or around large-scale construction land. Barriers were distributed mainly in Zhangqiu District and northern Licheng District. Based on these findings, hierarchical restoration areas were delimited. Differentiated development contradictions in restoration areas were discussed, and corresponding ecological protection measures were proposed. An ecological security optimization pattern of "one center, two wings, and two belts" was finally proposed to provide planning strategies for decision-makers.

Introduction to Stochastic Calculus and Its Applications

Stochastic Calculus deals with the rate of change of functions with respect to the randomness. It has been applied in the field of financial mathematics. When we look at the graph of stock prices, there are many irregular price fluctuations. The main tool for modeling the small fluctuations is the Brownian motion. This paper will mainly introduce the concept of Brownian motion, the concept of random walks, and the main flavors of stochastic calculus, Ito calculus. I will also discuss various applications, especially in the finance area.

Continuous time random walk concepts applied to extended mode coupling theory: a study of the Stokes–Einstein breakdown

In an attempt to extend the mode coupling theory (MCT) to lower temperatures, some years back an Unified theory was proposed which within the MCT framework incorporated the activated dynamics via the random first order transition theory (RFOT). The theory successfully showed that there is hopping induced diffusive dynamics and the modified MCT coupled to the activated motion continues till low temperatures. Here we show that the theory although successful in describing other properties of supercooled liquids is unable to capture the Stokes–Einstein breakdown. We then show using continuous time random work (CTRW) formalism that the Unified theory is equivalent to a CTRW dynamics in presence of two waiting time distributions. It is known from earlier work on CTRW that in such cases the total dynamics is dominated by the fast motion. This explains the failure of the Unified theory in predicting the SE breakdown as both the structural relaxation and the diffusion process are described by the comparatively fast MCT like dynamics. The study also predicts that other forms of extended MCT with Markovian hopping kernel will face a similar issue. We next modify the Unified theory by applying the concept of renewal theory, usually used in CTRW models where the distribution has a long tail. According to this theory the first jump given by the persistent time is slower than the subsequent jumps given by the exchange time. We first show that for systems with two waiting time distributions even when both the distributions are exponential the persistent time is larger than the exchange time. We also identify the persistent time with the slower activated process. The extended Unified theory can now explain the SE breakdown. In this extended theory at low temperatures the structural relaxation is described by the activated dynamics whereas the diffusion is primarily determined by the MCT like dynamics leading to a decoupling between them. We also calculate a dynamic lengthscale from the wavenumber dependence of the relaxation time. We find that this dynamic length scale grows faster than the static length scale.

A Simple Model of Distribution of Current Over Cathodes of Vacuum Circuit Breakers

There are several hundreds of spots operating simultaneously on cathodes of vacuum arcs in high-power vacuum circuit breakers. In this work, the spot distribution along the contact surface is simulated by means of an approach that is based on the concept of surface density of spots and represents a natural alternative to tracing individual spots. An equation governing the evolution of the surface density of the spots or, equivalently, the distribution of macroscopic (averaged over individual spots) current density over the cathode is obtained by generalizing the concept of random walk of a single cathode spot in low-current vacuum arcs. The model relies on empirical parameters characterizing individual spots (the diffusion coefficient of the random motion of cathode spots and the velocity of drift superimposed over the random motion), which may be taken from experiments with low-current arcs, and does not involve adjustable parameters. The model is simple and physically transparent and correctly reproduces the trends observed in the experiments under conditions where the cathode arc attachment is diffuse. The distribution of the macroscopic current density on the cathode, given by the model, represents the boundary condition that is required for existing numerical models of vacuum arcs in high-power vacuum circuit breakers.

A species-clustered splitting scheme for the integration of large-scale chemical kinetics using detailed mechanisms

In this study, a species-clustered integrator for chemical kinetics with large detailed mechanisms based on operator-splitting is presented. The ordinary differential equation (ODE) system of large-scale chemical kinetics is split into clusters of species by using graph partition methods which have been intensely studied in areas of model reduction, parameterization and coarse-graining, e.g., diffusion maps based on the concept of Markov random walk. The definition of the weight (similarity) matrix is application-dependent and follows from chemical kinetics. Each species cluster is integrated by the variable-coefficient ODE solver VODE. The theoretically expected speedup in computational efficiency is reproduced by numerical experiments on three zero-dimensional (0D) auto-ignition problems, considering detailed hydrocarbon/air combustion mechanisms at varying scales, from 53 species with 325 reactions of methane to 2115 species with 8157 reactions of n-hexadecane. Optimal clustering weighing both prediction accuracy (for ignition delay and equilibrium temperature) and computational efficiency is implied with the clustering number N=2 for the 53-species methane mechanism, N=4 for the 561-species n-heptane mechanism and N=8 for the 2115-species n-hexadecane mechanism.

A novel source location privacy preservation technique to achieve enhanced privacy and network lifetime in WSNs

In this paper, we propose a two-phase routing technique using multiple virtual sources to provide enhanced source location privacy in Wireless Sensor Networks (WSNs). We use the concept of escape-angle and random walks that is based on potential energy. The proposed method routes packets to the base station via different virtual sources located at various positions in the network. The key idea of this work is to exploit the excess energy available in the non-hotspot areas of the network to generate dispersive routes between source node and the virtual sources. This approach maximizes safety-period without hampering the network lifetime. We present mathematical models to estimate the overall energy expenditure that incurs at each node during Min Hop Routing phase (phase two). We then determine the remaining amount of energy which could be used for Stochastic and Diffusive Routing phase (phase one). Simulation results show that the proposed technique achieves improved safety-period without hampering the network lifetime.

Deposition of diesel exhaust particles in the human lungs: theoretical simulations and experimental data

Background: Based on numerous experimental and epidemiological studies diesel exhaust particles (DEPs) have been evaluated as serious health hazards, whose uptake by inhalation causes an increase of the relative risk for lung cancer. Due to this circumstance, experimental and theoretical knowledge on the transport and deposition behaviour of such particles in the respiratory tract has to be permanently improved. Here, respective support is provided by models predicting aggregate shape and transport with high accuracy. Methods: Aggregate shape was modeled by using the computer program AGGREGATE, where respective particles are constructed from equally sized spherules by application of the random walk concept. Intrapulmonary transport and deposition of the particulate mass were simulated for a stochastic lung architecture and under the assumption of mathematically describable deposition mechanisms (Brownian motion, inertial impaction, interception, gravitational settling). Computations were conducted for particles ranging in size from 50 to 250 nm. In addition, three different breathing scenarios (sitting, light exercise, heavy exercise) were considered. Results: According to the results of the deposition computations theoretically constructed DEP aggregates exhibit decreasing total and regional deposition with growing particle diameter. Depending on aggregate size and breathing conditions, total deposition ranges from 15.7% to 71.5%, whereas tubular deposition adopts values between 10.1% and 35.2%. Alveolar deposition may be quantified with 4.7% to 27.0%. Except for alveolar accumulation of aggregates ≤100 nm, deposition of DEPs negatively correlates with breathing intensity. Local deposition of such aggregates is commonly characterized by the development of respective modes in airway generations 20 to 22. Conclusions: The present study leads to the conclusion that DEPs may bear the potential for an increased deposition in those lung regions, which are highly susceptible for the development of lung cancer. Thereby, the amount of particulate mass accumulated in airways and alveoli depends on both particle size and the intensity of particle uptake by inhalation.

On the quantum CTRW approach

The concept of continuous-time random walk is generalized into the quantum approach using a completely positive map. This approach introduces in a phenomenological way the concept of disorder in the transport problem of a quantum open system. If the waiting-time of the continuous-time renewal approach is exponential we recover a semigroup for a dissipative quantum walk. Two models of non-Markovian evolution have been solved considering different types of waiting-time functions.

Evaluation of the Applicability of Random Walks for Generation of Material Flow Network Models

Modeling material flow systems as networks in terms of graphs is a straightforward approach to investigate these systems. In this context, the nodes represent the workstations and the edges indicate a material flow between two nodes. Furthermore, the individual edges are weighted by the amount of material flow that actually takes place between two workstations. The network model is simple but powerful, because it is easy to build and understand, but it also contains the main information required for the analysis of flow in manufacturing systems. Due to the lack of availability of material flow data from industry, it is sometimes necessary to replicate given real networks in order to perform network based research in manufacturing. The majority of available approaches in the literature generate a network by iteratively adding nodes until the desired network size is reached. In most cases the edge weight is not considered, thus there is no information about the intensity of material flow. However, in the context of manufacturing systems there is a movement of individual parts through the system. These parts have a specific order in which they have to be processed. It is therefore essential to simulate the movement of the parts as a whole. For this purpose, we use the concept of random walks. Random walks describe the movement from a random start node to a random end node, whereby the choice of the subsequent steps is also random. We modify the procedure of random walks so that the properties of these walks are based on the characteristics of real material flow networks, such as number of parts (number of random walks), average number of operations (number of nodes visited by a random walk), etc. In this paper, we show how random walks can be used to generate weighted networks, which exhibit similar properties like real material flow networks. Finally, we evaluate our approach by comparing the properties of real material flow networks and the networks generated using random walks.