Random Walk Path Research Articles

Random walks in the high-dimensional limit II: The crinkled subordinator

A crinkled subordinator is an ℓ2-valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other jumps. We show that the path of a d-dimensional random walk with n independent identically distributed steps with heavy-tailed distribution of the radial components and asymptotically orthogonal angular components converges in distribution in the Hausdorff distance up to isometry and also in the Gromov–Hausdorff sense, if viewed as a random metric space, to the closed range of a crinkled subordinator, as d,n→∞.

An ecological network for large carnivores as a key tool for protecting landscape connectivity in the Carpathians

ABSTRACT The main aim of our study was to delineate a coherent ecological network for large carnivores, using a unified methodology, which would be flexible and serve as a basis for nature and habitat connectivity protection across the Carpathians. We applied a comprehensive methodological approach based on habitat suitability modelling followed by connectivity modelling. For the habitat suitability modelling, we used a robust set of large carnivores occurrence data, which allowed us to build a solid model. From the model, we derived a layer of patches of suitable habitat of required quality and resistance surface to express landscape permeability. Both outputs were crucial for the connectivity modelling. We applied Random Walk and Least-Cost Path models and combined them to determine feasible connections between suitable habitats. With expert discussions and editing/improvement of modelled corridors, we developed a unique ecological network consisting of various components adapted to the heterogeneity of the Carpathian landscape.

Deformed random walk: Suppression of randomness and inhomogeneous diffusion.

We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the q algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for q→1 and a suppression of randomness is manifested for the DRW with -1<γ_{q}<1 and γ_{q}=1-q. The passage to the continuum of the master equation associated to the DRW led to a van Kampen inhomogeneous diffusion equation when the mobility and the temperature are proportional to 1+γ_{q}x, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at x=-1/γ_{q} consistent with the fixed point of the DRW. Complementarily, a comparison with the Plastino-Plastino Fokker-Planck equationis discussed. The two-dimensional case is also studied, by obtaining a 2D deformed random walk and its associated deformed 2D Fokker-Planck equation, which give place to a convergence of the 2D paths for -1<γ_{q_{1}},γ_{q_{2}}<1 and a diffusion with inhomogeneities controlled by two deformation parameters γ_{q_{1}},γ_{q_{2}} in the directions x and y. In both the one-dimensional and the two-dimensional cases, the transformation γ_{q}→-γ_{q} implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.

Self-supervised contrastive learning with random walks for medical image segmentation with limited annotations.

Medical image segmentation has seen significant progress through the use of supervised deep learning. Hereby, large annotated datasets were employed to reliably segment anatomical structures. To reduce the requirement for annotated training data, self-supervised pre-training strategies on non-annotated data were designed. Especially contrastive learning schemes operating on dense pixel-wise representations have been introduced as an effective tool. In this work, we expand on this strategy and leverage inherent anatomical similarities in medical imaging data. We apply our approach to the task of semantic segmentation in a semi-supervised setting with limited amounts of annotated volumes. Trained alongside a segmentation loss in one single training stage, a contrastive loss aids to differentiate between salient anatomical regions that conform to the available annotations. Our approach builds upon the work of Jabri et al. (2020), who proposed cyclical contrastive random walks (CCRW) for self-supervision on palindromes of video frames. We adapt this scheme to operate on entries of paired embedded image slices. Using paths of cyclical random walks bypasses the need for negative samples, as commonly used in contrastive approaches, enabling the algorithm to discriminate among relevant salient (anatomical) regions implicitly. Further, a multi-level supervision strategy is employed, ensuring adequate representations of local and global characteristics of anatomical structures. The effectiveness of reducing the amount of required annotations is shown on three MRI datasets. A median increase of 8.01 and 5.90 pp in the Dice Similarity Coefficient (DSC) compared to our baseline could be achieved across all three datasets in the case of one and two available annotated examples per dataset.

Random-walk-path solution of unsteady flow equations for general channel networks

The random walk path method (RWP) is a stochastic model that transforms the solution of the unsteady open channel flow equations into a probability problem. In this paper, this method is introduced to solve the discretized Saint-Venant equations, with probabilistic representation of the water levels at junctions. Different boundary conditions, which specify either water levels or discharge hydrographs, can be implemented by specifying how walkers react when arriving at the boundaries. The pointwise solution at river junctions can be obtained via random walk simulations. The efficacy of the model was verified against the following test cases: (1) a real-world looped channel example documented in the manual of the software Hec-Ras and (2) a hypothetical channel system consisting of dendritic and divergent networks. The number of random simulations is an important aspect of the RWP method and needs to be chosen by considering the trade-off between precision and computational efficiency. The impact of the boundary water level or discharge on the water levels at internal nodes can be quantitatively evaluated by adopting the terminal weights, which present a distinct advantage of the RWP method. This assessment of water levels can serve as a guide in the operation of hydraulic controls, such as dams, sluices and pumps, to effectively regulate the flow in mitigating flood risks.

Covid-19’s Fever on Financial Markets in China: The Interaction Between Foreign Exchange Market and the Stock Market

The impacts of the COVID-19 pandemic on economies are different, however financial markets followed a similar pattern. This paper investigates the preliminary effect of the pandemic on China’s stock exchange market and foreign exchange market by comparing the pre-pandemic and post-pandemic periods. The results indicate that there is a significant interaction between two markets and both markets do not behave randomly. This linkage is crucial to determine the macroeconomic policies against a random walk. We investigate that the monetary policy interventions have positive effect to increase efficiency of financial markets. Moreover, financial markets do not follow a random walk path. It can be said that now, China’s money market in an upward trend due to the relatively reduce in uncertainty and pessimist expectations.

Dynamic Compressive Data Gathering using Angle-based Random Walk in Hybrid WSNs

This paper addresses the problems of energy consumption, load-balancing, and Compressive Sensing (CS) recovery performance in hybrid proactive-reactive wireless sensor network, where sensors are categorized into two representative and relay nodes. We jointly consider the dynamic compressive data gathering, angle-based random walk, and spatial correlation to propose an efficient method named “Dynamic Compressive Data Gathering using Angle-based Random Walk (DCDG-ARW)”. The proposed scheme uses the CS theory along with the angle-based random walks to transmit CS measurements from sensors to the sink node. To select a starting node for each random walk path, a heuristic mechanism called Distance-and Correlation-aware Node Selection (DCNS) is presented. Furthermore, a transition probability is defined for the proposed angle-based random walks to jointly investigate dynamic changes of the network, increasing load-balancing, and improving the quality of the measurement matrix. The effectiveness of DCDG-ARW is evaluated in terms of energy consumption, load variance, network’s lifetime, and reconstruction error. Based on the simulation results, the proposed DCDG-ARW algorithm illustrates significant performance improvements in comparison to its counterparts across all the aforementioned aspects.

A network-based CNN model to identify the hidden information in text data

With the development of the internet and big data, the missing or hidden information identification of text data has become an imperative task. At present, the challenge in the hidden information study is judging whether there is hidden information and where it exists. In this paper, hidden information refers to the words that do not appear in a sentence, however, they have certain correlations with the existing words or sentence and have a great influence on the comprehension of a sentence or part of the text data. This paper focuses on discovering the key and influential hidden information in the text data. A keyword-based hidden information extraction framework is proposed in this paper to search hidden entities, with the assumption that the importance of hidden objects is reflected by the keywords in the text data. A network-based Convolution Neural Network (CNN) model is developed to identify the hidden information related to keywords. The model is based on the results of CNN, and cosine similarity is used to judge whether there is hidden information in the source text data or not. We primarily form the word co-occurrence network of text, select the words with the highest degree as keywords, and generate random walk paths on the network. Besides, we use the random walk path where the last word is the keyword to train CNN. In the experimental section, the proposed model is applied to the dataset in 20Newgroups. The results show that the proposed model can effectively identify the hidden information associated with the keywords in the source text data, and the detection accuracy of keywords can reach 98%–99% achieved by CNN.

{text{COSNet}}_i: ComplexOme-Structural Network Interpreter used to study spatial enrichment in metazoan ribosomes

BackgroundUpon environmental stimuli, ribosomes are surmised to undergo compositional rearrangements due to abundance changes among proteins assembled into the complex, leading to modulated structural and functional characteristics. Here, we present the ComplexOme-Structural Network Interpreter ({text{COSNet}}_i), a computational method to allow testing whether ribosomal proteins (rProteins) that exhibit abundance changes under specific conditions are spatially confined to particular regions within the large ribosomal complex.Results{text{COSNet}}_i translates experimentally determined structures into graphs, with nodes representing proteins and edges the spatial proximity between them. In its first implementation, {text{COSNet}}_i considers rProteins and ignores rRNA and other objects. Spatial regions are defined using a random walk with restart methodology, followed by a procedure to obtain a minimum set of regions that cover all proteins in the complex. Structural coherence is achieved by applying weights to the edges reflecting the physical proximity between purportedly contacting proteins. The weighting probabilistically guides the random-walk path trajectory. Parameter tuning during region selection provides the option to tailor the method to specific biological questions by yielding regions of different sizes with minimum overlaps. In addition, other graph community detection algorithms may be used for the {text{COSNet}}_i workflow, considering that they yield different sized, non-overlapping regions. All tested algorithms result in the same node kernels under equivalent regions. Based on the defined regions, available abundance change information of proteins is mapped onto the graph and subsequently tested for enrichment in any of the defined spatial regions. We applied {text{COSNet}}_i to the cytosolic ribosome structures of Saccharomyces cerevisiae, Oryctolagus cuniculus, and Triticum aestivum using datasets with available quantitative protein abundance change information. We found that in yeast, substoichiometric rProteins depleted from translating polysomes are significantly constrained to a ribosomal region close to the tRNA entry and exit sites.Conclusions{text{COSNet}}_i offers a computational method to partition multi-protein complexes into structural regions and a statistical approach to test for spatial enrichments of any given subsets of proteins. {text{COSNet}}_i is applicable to any multi-protein complex given appropriate structural and abundance-change data. {text{COSNet}}_i is publicly available as a GitHub repository https://github.com/MSeidelFed/COSNet_i and can be installed using the python installer pip.

Airline transportation and arrival time of international disease spread: A case study of Covid-19.

In this era of globalization, airline transportation has greatly increased international trade and travel within the World Airport Network (WAN). Unfortunately, this convenience has expanded the scope of infectious disease spread from a local to a worldwide occurrence. Thus, scholars have proposed several methods to measure the distances between airports and define the relationship between the distances and arrival times of infectious diseases in various countries. However, such studies suffer from the following limitations. (1) Only traditional statistical methods or graphical representations were utilized to show that the effective distance performed better than the geographical distance technique. Researchers seldom use the survival model to quantify the actual differences among arrival times via various distance methods. (2) Although scholars have found that most diseases tend to spread via the random walk rather than the shortest path method, this hypothesis may no longer be true because the network has been severally altered due to recent COVID-related travel reductions. Therefore, we used 2017 IATA (International Air Transport Association) to establish an airline network via various chosen path strategies (random walk and shortest path). Then, we employed these two networks to quantify each model’s predictive performance in order to estimate the importation probability function of COVID-19 into various countries. The effective distance model was found to more accurately predict arrival dates of COVID-19 than the geographical distance model. However, if pre-Covid airline data is included, the path of disease spread might not follow the random walk theory due to recent flight suspensions and travel restrictions during the epidemic. Lastly, when testing effective distance, the inverse distance survival model and the Cox model yielded very similar importation risk estimates. The results can help authorities design more effective international epidemic prevention and control strategies.

Magnetic field line random walk and solar energetic particle path lengths

Context. In 2020 May-June, six solar energetic ion events were observed by the Parker Solar Probe/IS⊙IS instrument suite at ≈0.35 AU from the Sun. From standard velocity-dispersion analysis, the apparent ion path length is ≈0.625 AU at the onset of each event. Aims. We develop a formalism for estimating the path length of random-walking magnetic field lines to explain why the apparent ion path length at an event onset greatly exceeds the radial distance from the Sun for these events. Methods. We developed analytical estimates of the average increase in path length of random-walking magnetic field lines, relative to the unperturbed mean field. Monte Carlo simulations of field line and particle trajectories in a model of solar wind turbulence were used to validate the formalism and study the path lengths of particle guiding-center and full-orbital trajectories. The formalism was implemented in a global solar wind model, and the results are compared with ion path lengths inferred from IS⊙IS observations. Results. Both a simple estimate and a rigorous theoretical formulation are obtained for field-lines’ path length increase as a function of path length along the large-scale field. From simulated field line and particle trajectories, we find that particle guiding centers can have path lengths somewhat shorter than the average field line path length, while particle orbits can have substantially longer path lengths due to their gyromotion with a nonzero effective pitch angle. Conclusions. The long apparent path length during these solar energetic ion events can be explained by (1) a magnetic field line path length increase due to the field line random walk and (2) particle transport about the guiding center with a nonzero effective pitch angle due to pitch angle scattering. Our formalism for computing the magnetic field line path length, accounting for turbulent fluctuations, may be useful for application to solar particle transport in general.

Magnetic Field Line Random Walk and Solar Energetic Particle Path Lengths: Stochastic Theory and PSP/ISoIS Observation

We thank Junxiang Hu for useful discussions. This research is partially supported by the Parker Solar Probe mission and the IS IS project (contract NNN06AA01C) and a subcontract to University of Delaware from Princeton University (SUB0000165). Additional support is acknowledged from the NASA Living With a Star (LWS) program (NNX17AB79G) and HSR program (80NSSC18K1210 & 80NSSC18K1648) and Thailand Science Research and Innovation (RTA6280002). The IS IS data and visualization tools are available to the community at https://spacephysics.princeton.edu/missions-instruments/isois; data are also available via the NASA Space Physics Data Facility. PSP was designed, built, and is now operated by the Johns Hopkins Applied Physics Laboratory as part of NASA’s LWS program (contract NNN06AA01C). Support from the LWS management and technical team has played a critical role in the success of the PSP mission.

A Comparative Study of Graph Kernels and Clustering Algorithms

Graph kernels have evolved as a promising and popular method for graph clustering over the last decade. In this work, comparative study on the five standard graph kernel techniques for graph clustering has been performed. The graph kernels, namely vertex histogram kernel, shortest path kernel, graphlet kernel, k-step random walk kernel, and Weisfeiler-Lehman kernel have been compared for graph clustering. The clustering methods considered for the kernel comparison are hierarchical, k-means, model-based, fuzzy-based, and self-organizing map clustering techniques. The comparative study of kernel methods over the clustering techniques is performed on MUTAG benchmark dataset. Clustering performance is assessed with internal validation performance parameters such as connectivity, Dunn, and the silhouette index. Finally, the comparative analysis is done to facilitate researchers for selecting the appropriate kernel method for effective graph clustering. The proposed methodology elicits k-step random walk and shortest path kernel have performed best among all graph clustering approaches.

Density Sensitive Random Walk for Local Community Detection

Given a network, local community detection aims at finding the community that contains a set of query nodes (seed nodes). Random walk (RW) based algorithms have shown great success in various local community detection scenarios. Starting from the seed nodes, RW based algorithms continuously sample random walk paths to get the clustering result. However, current RW based algorithms for local clustering are faced with the following two problems. The random walker is insensitive to the community boundary and might have an unbalanced walk. These problems would have a negative effect on the clustering result of RW based algorithms. In this paper, we propose a density sensitive random walk algorithms (DSRW) for local community detection. By integrating the graph density information into the random walk process, the problems are resolved and the clustering result is improved. We provide the convergence proof of DSRW algorithm and perform extensive experiments on both real and synthetic datasets. Results show that our DSRW algorithm has achieved the state-of-the-art result in most scenarios.

Walk-Steered Convolution for Graph Classification.

Graph classification is a fundamental but challenging issue for numerous real-world applications. Despite recent great progress in image/video classification, convolutional neural networks (CNNs) cannot yet cater to graphs well because of graphical non-Euclidean topology. In this article, we propose a walk-steered convolutional (WSC) network to assemble the essential success of standard CNNs, as well as the powerful representation ability of random walk. Instead of deterministic neighbor searching used in previous graphical CNNs, we construct multiscale walk fields (a.k.a. local receptive fields) with random walk paths to depict subgraph structures and advocate graph scalability. To express the internal variations of a walk field, Gaussian mixture models are introduced to encode the principal components of walk paths therein. As an analogy to a standard convolution kernel on image, Gaussian models implicitly coordinate those unordered vertices/nodes and edges in a local receptive field after projecting to the gradient space of Gaussian parameters. We further stack graph coarsening upon Gaussian encoding by using dynamic clustering, such that high-level semantics of graph can be well learned like the conventional pooling on image. The experimental results on several public data sets demonstrate the superiority of our proposed WSC method over many state of the arts for graph classification.

On moments of Brownian functionals and their interpretation in terms of random walks

An identity in law, shown in Mansuy and Yor (2008); Yor (1991), involves integrals of quadratic functionals of the Brownian motion. The corresponding equalities of moments bring in equalities of multiple integrals of joint moments of the Brownian motion taken at different times. In the present paper, we compute these moments, and deduce an interpretation of some of these computations in terms of combinatorial sums indexed by different paths of random walks.

Randomness for Asset Prices Constrained by Price Limit Regimes: A Malaysian Case Study

Empirical works testing randomness of stock prices are abundant. Such findings, however, can be challenged if time-series datasets examined are subjected to price limits, which, ex vi termini, enforce bounded movements. This article examines the random walk hypothesis for the Malaysian equity market under three price limit regimes from January 1994 to September 2017. Price limits have been active in Malaysia’s solitary bourse since 1989 with periodic revisions. Identifying a sample of 407 actively traded instruments that triggered limits on 5,843 occasions, the authors use parametric (Ljung–Box, Lo and MacKinlay, and Chow–Denning) and nonparametric (Wald–Wolfowitz runs and Broock–Decher–Scheinkman independence) tests to investigate whether prices under different circuit breaker regimes follow a random walk path, an indicator of market efficiency. Upon comparison with the composite FBMKLCI index and sectoral indexes—the bulk of which reject the random walk hypothesis—the study finds considerable support for randomness across all regimes for upper and lower limit-hit stocks. Moreover, progressive tightening of the price limit appears to correspond with a lower proportion of limit-hit stocks following a random path. The findings carry implications for regulators and academia. First, the unusually wide price band in Malaysia appears to outperform the tighter limits studied earlier. Second, the findings furnish direct evidence of price randomness and price discovery in the financial economics literature and provide indirect evidence of circuit breaker efficacy to market microstructure literature. Third, econometric issues arising from disparate results of various tests are indicated, with ramifications for methodological developments. The article concludes with suggestions for future research. TOPICS:Private equity, exchanges/markets/clearinghouses, emerging, statistical methods

Estimating graph parameters with random walks

An algorithm observes the trajectories of random walks over an unknown graph G , starting from the same vertex x , as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges m up to a bounded factor in O(t_{\mathrm {rel}}^{3/4}\sqrt{m/d}\,) steps, where t_{\mathrm {rel}} is the relaxation time of the lazy random walk on G and d is the minimum degree in G . Alternatively, m can be estimated in O(t_{\mathrm {unif}} + t_{\mathrm {rel}}^{5/6}\sqrt{n}\,) , where n is the number of vertices and t_{\mathrm {unif}} is the uniform mixing time on G . The number of vertices n can then be estimated up to a bounded factor in an additional O(t_{\mathrm {unif}}\frac{m}{n}) steps. Our algorithms are based on counting the number of intersections of random walk paths X,Y , i.e. the number of pairs (t,s) such that X_t=Y_s . This improves on previous estimates which only consider collisions (i.e. times t with X_t=Y_t ). We also show that the complexity of our algorithms is optimal, even when restricting to graphs with a prescribed relaxation time. Finally, we show that, given either m or the mixing time of G , we can compute the “other parameter” with a self-stopping algorithm.

Link Prediction Based on Deep Convolutional Neural Network

In recent years, endless link prediction algorithms based on network representation learning have emerged. Network representation learning mainly constructs feature vectors by capturing the neighborhood structure information of network nodes for link prediction. However, this type of algorithm only focuses on learning topology information from the simple neighbor network node. For example, DeepWalk takes a random walk path as the neighborhood of nodes. In addition, such algorithms only take advantage of the potential features of nodes, but the explicit features of nodes play a good role in link prediction. In this paper, a link prediction method based on deep convolutional neural network is proposed. It constructs a model of the residual attention network to capture the link structure features from the sub-graph. Further study finds that the information flow transmission efficiency of the residual attention mechanism was not high, so a densely convolutional neural network model was proposed for link prediction. We evaluate our proposed method on four published data sets. The results show that our method is better than several other benchmark algorithms on link prediction.

Finitary random interlacements and the Gaboriau–Lyons problem

The von Neumann–Day problem asks whether every non-amenable group contains a non-abelian free group. It was answered in the negative by Ol’shanskii in the 1980s. The measurable version (formulated by Gaboriau–Lyons) asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau–Lyons. The proof uses an approximation to the random interlacement process by random multisets of geometrically-killed random walk paths. There are two applications: (1) the Gaboriau–Lyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all Bernoulli shifts factor onto each other.