Random Walk Framework Research Articles

Continuous Time Random Walk and Migration–Proliferation Dichotomy of Brain Cancer

A theory of fractional kinetics of glial cancer cells is presented. A role of the migration-proliferation dichotomy in the fractional cancer cell dynamics in the outer-invasive zone is discussed and explained in the framework of a continuous time random walk. The main suggested model is based on a construction of a 3D comb model, where the migration-proliferation dichotomy becomes naturally apparent and the outer-invasive zone of glioma cancer is considered as a fractal composite with a fractal dimension Dfr< 3.

Weak Ergodicity Breaking of Membrane Receptor Motion Stemming from Random Diffusivity

The application of techniques such as fluorescence correlation spectroscopy and single particle tracking to the plasma membrane of living cells continuously evidences subdiffusive motion of proteins and lipids. This anomalous motion is generally associated to the interplay of molecular crowding, diffusion barriers and specific interactions. Addressing the cause of subdiffusion is essential for understanding molecular mechanisms underlying cellular function, such as target search, kinetics of transport-limited reactions, trafficking and signalling. Recently, the subdiffusive motion of some cellular and membrane components has been associated to weak ergodicity breaking and attributed to transient immobilization caused by interactions with microtubules or actin cytoskeleton. These works have opened new questions about the role of nonergodic subdiffusion in the dynamics of living systems.By means of single particle tracking experiments, we show that the dynamics of DC-SIGN, a transmembrane receptor with unique pathogen recognition capabilities, also reveals subdiffusion with signatures of weak ergodicity breaking and aging. In contrast to other biological systems, we demonstrate that DC-SIGN nonergodic subdiffusion cannot be explained by transient immobilization, but rather is compatible with dynamical heterogeneity induced by spatiotemporal changes of diffusivity. While nonergodic diffusion due to long-lived immobilization events can be understood within the framework of continuous-time random walk, our experimental data are accurately interpreted through a new theoretical model describing anomalous transport in biological systems and complex media. A comprehensive analysis of mutated forms of the receptor allowed us to establish a link between receptor molecular structure, nonergodic diffusion and function. Our results underscore the role of spatiotemporal disorder in the dynamics of cell membrane receptors. In addition, they have broad implications to other heterogeneous systems where the occurrence of nonergodicity remains unexplored.

Maximum therapeutic effect of glioma treatment by radio-frequency electric field

The influence of a radio-frequency electric field on glioma brain cancer development is considered. Specifically, the effectiveness of this medical technology against invasive cells with a high motility, when switching between migrating and proliferating phenotypes takes place, is addressed. It is shown that glioma development under the external field treatment can be modeled in the framework of the continuous time random walk, where the segregation of proliferating from not proliferating cells by the external electric field is naturally accounted for. The constructed reaction-transport equation is based on the fact that cytokinesis affects the cell dynamics strongly, and the cell kinetics is non-Markovian, which is reflected in both the transport and reaction properties of cells. As a result of the interplay between cell proliferation and cell degradation, migrating cancer cells spread freely without being influenced by the details of the degradation of the proliferating cells, which leads to a decrease in the effectiveness of the treatment.

Fractional kinetics under external forcing

Fractional tumor development is considered in the framework of one-dimensional continuous time random walks (CTRW) in the presence of chemotherapy. The chemotherapy influence on the CTRW is studied by observations of both stationary solutions due to proliferation and fractional evolution in time.

Contents and time sensitive document ranking of scientific literature

Abstract A new link-based document ranking framework is devised with at its heart, a contents and time sensitive random literature explorer designed to more accurately model the behaviour of readers of scientific documents. In particular, our ranking framework dynamically adjusts its random walk parameters according to both contents and age of encountered documents, thus incorporating the diversity of topics and how they evolve over time into the score of a scientific publication. Our random walk framework results in a ranking of scientific documents which is shown to be more effective in facilitating literature exploration than PageRank measured against a proxy gold standard based on papers’ potential usefulness in facilitating later research. One of its many strengths lies in its practical value in reliably retrieving and placing promisingly useful papers at the top of its ranking.

Survival of Classical and Quantum Particles in the Presence of Traps

We present a detailed comparison of the motion of a classical and of a quantum particle in the presence of trapping sites, within the framework of continuous-time classical and quantum random walk. The main emphasis is on the qualitative differences in the temporal behavior of the survival probabilities of both kinds of particles. As a general rule, static traps are far less efficient to absorb quantum particles than classical ones. Several lattice geometries are successively considered: an infinite chain with a single trap, a finite ring with a single trap, a finite ring with several traps, and an infinite chain and a higher-dimensional lattice with a random distribution of traps with a given density. For the latter disordered systems, the classical and the quantum survival probabilities obey a stretched exponential asymptotic decay, albeit with different exponents. These results confirm earlier predictions, and the corresponding amplitudes are evaluated. In the one-dimensional geometry of the infinite chain, we obtain a full analytical prediction for the amplitude of the quantum problem, including its dependence on the trap density and strength.

Natural Organic Matter Transport Modeling with a Continuous Time Random Walk Approach

In transport experiments through columns packed with naturally Fe/Al oxide-coated quartz sand, breakthrough curves (BTCs) of natural organic matter (NOM) displayed strong and persistent power law tailing that could not be described by the classical advection-dispersion equation. Tailing was not observed in BTCs for a nonreactive tracer (sulforhodamine B); therefore, the anomalous transport is attributed to diverse adsorptive behavior of the polydisperse NOM sample rather than to physical heterogeneity of the porous medium. NOM BTC tailing became more pronounced with decreases in pH and increases in ionic strength, conditions previously shown to be associated with enhanced preferential adsorption of intermediate to high molecular weight NOM components. Drawing from previous work on anomalous solute transport, we develop an approach to model NOM transport within the framework of a continuous time random walk (CTRW) and show that under all conditions examined, the CTRW model is able to capture tailing of NOM BTCs by accounting for differences in transport rates of NOM fractions through a distribution of effective retardation factors. These results demonstrate the importance of considering effects of adsorptive fractionation on NOM mobility, and illustrate the ability of the CTRW model to describe transport of a multicomponent solute.

Origins of anomalous transport in heterogeneous media: Structural and dynamic controls

Anomalous (or “non-Fickian”) transport is ubiquitous in the context of tracer migration in geological formations. We quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous porous medium under uniform (in the mean) flow conditions; we focus on anomalous transport which arises in the complex flow patterns of lognormally distributed hydraulic conductivity (K) fields, with several decades of K values. Transport in the domains is determined by a particle tracking technique and characterized by breakthrough curves (BTCs). The BTC averaged over multiple realizations demonstrates anomalous transport in all cases, which is accounted for entirely by a power law distribution ∼t−1−β of local transition times. The latter is contained in the probability density function ψ(t) of transition times, embedded in the framework of a continuous time random walk (CTRW). A unique feature of our analysis is the derivation of ψ(t) as a function of parameters quantifying the heterogeneity of the domain. In this context, we first establish the dominance of preferential pathways across each domain, and characterize the statistics of these pathways by forming a particle-visitation weighted histogram, Hw(K), of the hydraulic conductivity. By converting the ln(K) dependence of Hw(K) into time, we demonstrate the equivalence of Hw(K) and ψ(t), and delineate the region of Hw(K) that forms the power law of ψ(t). This thus defines the origin of anomalous transport. Analysis of the preferential pathways clearly demonstrates the limitations of critical path analysis and percolation theory as a basis for determining the origin of anomalous transport. Furthermore, we derive an expression defining the power law exponent β in terms of the Hw(K) parameters. The equivalence between Hw(K) and ψ(t) is a remarkable result, particularly given the nature of the K heterogeneity, the complexity of the flow field within each realization, and the statistics of the particle transitions.

Lattice-free descriptions of collective motion with crowding and adhesion

Cell-to-cell adhesion is an important aspect of malignant spreading that is often observed in images from the experimental cell biology literature. Since cell-to-cell adhesion plays an important role in controlling the movement of individual malignant cells, it is likely that cell-to-cell adhesion also influences the spatial spreading of populations of such cells. Therefore, it is important for us to develop biologically realistic simulation tools that can mimic the key features of such collective spreading processes to improve our understanding of how cell-to-cell adhesion influences the spreading of cell populations. Previous models of collective cell spreading with adhesion have used lattice-based random walk frameworks which may lead to unrealistic results, since the agents in the random walk simulations always move across an artificial underlying lattice structure. This is particularly problematic in high-density regions where it is clear that agents in the random walk align along the underlying lattice, whereas no such regular alignment is ever observed experimentally. To address these limitations, we present a lattice-free model of collective cell migration that explicitly incorporates crowding and adhesion. We derive a partial differential equation description of the discrete process and show that averaged simulation results compare very well with numerical solutions of the partial differential equation.

Multi-Scale Kernels Using Random Walks

We introduce novel multi-scale kernels using the random walk framework and derive corresponding embeddings and pairwise distances. The fractional moments of the rate of continuous time random walk (equivalently diffusion rate) are used to discover higher order kernels (or similarities) between pair of points. The formulated kernels are isometry, scale and tessellation invariant, can be made globally or locally shape aware and are insensitive to partial objects and noise based on the moment and influence parameters. In addition, the corresponding kernel distances and embeddings are convergent and efficiently computable. We introduce dual Green's mean signatures based on the kernels and discuss the applicability of the multi-scale distance and embedding. Collectively, we present a unified view of popular embeddings and distance metrics while recovering intuitive probabilistic interpretations on discrete surface meshes.

Structural disorder and anomalous diffusion in random packing of spheres.

Nowadays Nuclear Magnetic Resonance diffusion (dNMR) measurements of water molecules in heterogeneous systems have broad applications in material science, biophysics and medicine. Up to now, microstructural rearrangement in media has been experimentally investigated by studying the diffusion coefficient (D(t)) behavior in the tortuosity limit. However, this method is not able to describe structural disorder and transitions in complex systems. Here we show that, according to the continuous time random walk framework, the dNMR measurable parameter α, quantifying the anomalous regime of D(t), provides a quantitative characterization of structural disorder and structural transition in heterogeneous systems. To demonstrate this, we compare α measurements obtained in random packed monodisperse micro-spheres with Molecular Dynamics simulations of disordered porous media and 3D Monte Carlo simulation of particles diffusion in these kind of systems. Experimental results agree well with simulations that correlate the most used parameters and functions characterizing the disorder in porous media.

Diagnosing Heterogeneous Dynamics in Single-Molecule/Particle Trajectories with Multiscale Wavelets

We describe a simple automated method to extract and quantify transient heterogeneous dynamical changes from large data sets generated in single-molecule/particle tracking experiments. Based on wavelet transform, the method transforms raw data to locally match dynamics of interest. This is accomplished using statistically adaptive universal thresholding, whose advantage is to avoid a single arbitrary threshold that might conceal individual variability across populations. How to implement this multiscale method is described, focusing on local confined diffusion separated by transient transport periods or hopping events, with three specific examples: in cell biology, biotechnology, and glassy colloid dynamics. The discussion is generalized within the framework of continuous time random walk. This computationally efficient method can run routinely on hundreds of millions of data points analyzed within an hour on a desktop personal computer.

Predicting Intraday Price Distributions at High Frequencies

We propose extensions to the continuous-time random walk (CTRW) framework so far mainly developed within the econophysics community. Using numerical methods, we extend the CTRW framework with more general marginal distributions than previously proposed. There are two main findings in this respect: First, modeling the returns with a Student-t distribution gives at least as good price distribution predictions as the other previously proposed distributions in this framework. Second, a mixed-Weibull waiting time distribution fits exceptionally well to our three-week long Nasdaq OMX data. When combined with standard financial econometric GARCH and ACD models and intraday seasonality filtering procedures new to the CTRW framework, our models deliver realistic intraday Value-at-Risk and Expected Shortfall predictions. Compared to the basic CTRW model, the total average performance improvement is typically around 70 percent. The effect of filtering out memory in waiting times is particularly noticeable: around 50 percent of the total performance improvement. Overall, our extensions and filtering methods make the CTRW framework a useful tool for intraday risk management (trading), especially at high frequencies of less than a minute or so.

Dynamics in Nonlinear Schrödinger Equation with dc bias: From Subdiffusion to Painlevé Transcendent

Dynamics of the nonlinear Schrödinger equation in the presence of a constant electric field is studied. Both discrete and continuous limits of the model are considered. For the discrete limit, a probabilistic description of subdiffusion is suggested and a subdiffusive spreading of a wave packet is explained in the framework of a continuous time random walk. In the continuous limit, the biased nonlinear Schrödinger equation is shown to be integrable, and solutions in the form of the Painlevé transcendents are obtained.

A modified random walk framework for handling negative ratings and generating explanations

The concept of random walk (RW) has been widely applied in the design of recommendation systems. RW-based approaches are effective in handling locality problem and taking extra information, such as the relationships between items or users, into consideration. However, the traditional RW-based approach has a serious limitation in handling bidirectional opinions. The propagation of positive and negative information simultaneously in a graph is nontrivial using random walk. To address the problem, this article presents a novel and efficient RW-based model that can handle both positive and negative comments with the guarantee of convergence. Furthermore, we argue that a good recommendation system should provide users not only a list of recommended items but also reasonable explanations for the decisions. Therefore, we propose a technique that generates explanations by backtracking the influential paths and subgraphs. The results of experiments on the MovieLens and Netflix datasets show that our model significantly outperforms state-of-the-art RW-based algorithms, and is capable of improving the overall performance in the ensemble with other models.

Iterative Self-Organizing Atherosclerotic Tissue Labeling in Intravascular Ultrasound Images and Comparison With Virtual Histology

Intravascular ultrasound (IVUS) is the predominant imaging modality in the field of interventional cardiology that provides real-time cross-sectional images of coronary arteries and the extent of atherosclerosis. Due to heterogeneity of lesions and stringent spatial/spectral behavior of tissues, atherosclerotic plaque characterization has always been a challenge and still is an open problem. In this paper, we present a systematic framework from in vitro data collection, histology preparation, IVUS-histology registration along with matching procedure, and finally a robust texture-derived unsupervised atherosclerotic plaque labeling. We have performed our algorithm on in vitro and in vivo images acquired with single-element 40 MHz and 64-elements phased array 20 MHz transducers, respectively. In former case, we have quantified results by local contrasting of constructed tissue colormaps with corresponding histology images employing an independent expert and in the latter case, virtual histology images have been utilized for comparison. We tackle one of the main challenges in the field that is the reliability of tissues behind arc of calcified plaques and validate the results through a novel random walks framework by incorporating underlying physics of ultrasound imaging. We conclude that proposed framework is a formidable approach for retrieving imperative information regarding tissues and building a reliable training dataset for supervised classification and its extension for in vivo applications.

Ultrasound confidence maps using random walks

Advances in ultrasound system development have led to a substantial improvement of image quality and to an increased use of ultrasound in clinical practice. Nevertheless, ultrasound attenuation and shadowing artifacts cannot be entirely avoided and continue to challenge medical image computing algorithms. We introduce a method for estimating a per-pixel confidence in the information depicted by ultrasound images, referred to as an ultrasound confidence map, which emphasizes uncertainty in attenuated and/or shadow regions. Our main novelty is the modeling of the confidence estimation problem within a random walks framework by taking into account ultrasound specific constraints. The solution to the random walks equilibrium problem is global and takes the entire image content into account. As a result, our method is applicable to a variety of ultrasound image acquisition setups. We demonstrate the applicability of our confidence maps for ultrasound shadow detection, 3D freehand ultrasound reconstruction, and multi-modal image registration.

RankCompete: Simultaneous ranking and clustering of information networks

Random walk was first introduced by Karl Pearson in 1905 and has inspired many research works in different fields. In recent years, random walk has been adopted in information network research, for example, ranking and similarity estimation. In this paper, we introduce a new model called RankCompete, which allows multiple random walkers in the same network (existing work mostly focus on random walks of a single category). By introducing the “competition” concept into the random walk framework, our method can fulfill both clustering and ranking tasks simultaneously and thus provide an effective analysis tool for networks. Compared with the traditional network ranking approaches, our new method focuses more on the structure of specialized clusters. Compared with the traditional graph clustering approaches, our new method provides a faster and more intuitive way to group the network nodes. We validate our approach on both bibliography networks and visual information networks, and the results show that our approach can obtain 100% perfect clustering results in clustering the DBLP 20 conferences and outperform the state-of-the-art on the Cora dataset. Furthermore, we show that our method can be effectively used to summarize personal photo collections.

Drug–target interaction prediction by random walk on the heterogeneous network

Predicting potential drug-target interactions from heterogeneous biological data is critical not only for better understanding of the various interactions and biological processes, but also for the development of novel drugs and the improvement of human medicines. In this paper, the method of Network-based Random Walk with Restart on the Heterogeneous network (NRWRH) is developed to predict potential drug-target interactions on a large scale under the hypothesis that similar drugs often target similar target proteins and the framework of Random Walk. Compared with traditional supervised or semi-supervised methods, NRWRH makes full use of the tool of the network for data integration to predict drug-target associations. It integrates three different networks (protein-protein similarity network, drug-drug similarity network, and known drug-target interaction networks) into a heterogeneous network by known drug-target interactions and implements the random walk on this heterogeneous network. When applied to four classes of important drug-target interactions including enzymes, ion channels, GPCRs and nuclear receptors, NRWRH significantly improves previous methods in terms of cross-validation and potential drug-target interaction prediction. Excellent performance enables us to suggest a number of new potential drug-target interactions for drug development.

Theoretical Simulation on the Chromatographic System Based on the Random Diffusion of the Separating Particles

In order to dynamically track the trajectory of diffusing molecules in a chromatography system, and to thoroughly understand its influence on chromatographic dynamics, we have developed software based on the framework of random walks in a confined space, with which the diffusion processes have been simulated. The influence of the filling rates, the form of the stationary phase, and the column length of a packed column on the chromatographic dynamics have been discussed based on these simulation results. It was concluded that shorter column lengths and larger filling rates result in a higher column efficiency. The particles to be separated normally show basic diffusion characteristics in the confined space. However, their flow behavior will increase with increasing external pressure. The simulation results indicate that the influence of the filling rate of the stationary phase and the column length on chromatographic dynamic behavior is similar to those seen in experiment, whereas the form of the stationary phase only has a slight effect because of the same close-packed barrier arrangement. This simulation method we proposed has some significance for the development of high-performance chromatography and novel separation technologies.