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Dimensional Interpolation for Random Walk
In this work, we employ a simple and accurate dimensional interpolation formula for the shapes of random walks at D = 3 and D = 2 based on the analytically known solutions at both limits D = ∞ and D = 1. The results obtained for the radius of gyration of an arbitrary shaped object have about 2% error compared with accurate numerical results at D = 3 and D = 2. We also calculated the asphericity for a three-dimensional random walk using the dimensional interpolation formula. The results agree very well with the numerically simulated results. The method is general and can be used to estimate other properties of random walks.
C-SAW: a framework for graph sampling and random walk on GPUs
Many applications require to learn, mine, analyze and visualize large-scale graphs. These graphs are often too large to be addressed efficiently using conventional graph processing technologies. Fortunately, recent research efforts find out graph sampling and random walk, which significantly reduce the size of original graphs, can benefit the tasks of learning, mining, analyzing and visualizing large graphs by capturing the desirable graph properties. This paper introduces C-SAW, the first framework that accelerates Sampling and Random Walk framework on GPUs. Particularly, C-SAW makes three contributions: First, our framework provides a generic API which allows users to implement a wide range of sampling and random walk algorithms with ease. Second, offloading this framework on GPU, we introduce warp-centric parallel selection, and two novel optimizations for collision migration. Third, towards supporting graphs that exceed the GPU memory capacity, we introduce efficient data transfer optimizations for out-of-memory and multi-GPU sampling, such as workload-aware scheduling and batched multi-instance sampling. Taken together, our framework constantly outperforms the state of the art projects in addition to the capability of supporting a wide range of sampling and random walk algorithms.
Neuromorphic scaling advantages for energy-efficient random walk computations
Neuromorphic computing, which aims to replicate the computational structure and architecture of the brain in synthetic hardware, has typically focused on artificial intelligence applications. What is less explored is whether such brain-inspired hardware can provide value beyond cognitive tasks. Here we show that the high degree of parallelism and configurability of spiking neuromorphic architectures makes them well suited to implement random walks via discrete-time Markov chains. Overall, these random walks are useful in Monte Carlo methods, which represent a fundamental computational tool for solving a wide range of numerical computing tasks. Using IBM’s TrueNorth and Intel’s Loihi neuromorphic computing platforms, we show that our neuromorphic computing algorithm for generating random walk approximations of diffusion offers advantages in energy-efficient computation compared with conventional approaches. We also show that our neuromorphic computing algorithm can be extended to more sophisticated jump-diffusion processes that are useful in a range of applications, including financial economics, particle physics and machine learning.
Statistical properties of sites visited by independent random walks
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. Here, we consider two independent random walks on hyper-cubic lattices and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0, t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the ordering probabilities in spatial dimensions d = 1, 2, 3, 4. We also study the time evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as a 1 ln t with a 1 = 0.970 508 in one dimension and as (ln t) 2 in two dimensions.
Understanding random-walk dynamical phase coexistence through waiting times
We study the appearance of first-order dynamical phase transitions (DPTs) as “intermittent” coexisting phases in the fluctuations of random walks on graphs. We show that the diverging timescale leading to critical behavior is the waiting time to jump from one phase to another. This timescale is crucial for observing the system's relaxation to stationarity and demonstrate ergodicity of the system at criticality. We illustrate these results through three analytical examples which provide insights into random walks exploring random graphs. Published by the American Physical Society 2024
Hypergraph Random Walks, Laplacians, and Clustering
We propose a flexible framework for clustering hypergraph-structured data based on recently proposed random walks utilizing edge-dependent vertex weights. When incorporating edge-dependent vertex weights (EDVW), a weight is associated with each vertex-hyperedge pair, yielding a weighted incidence matrix of the hypergraph. Such weightings have been utilized in term-document representations of text data sets. We explain how random walks with EDVW serve to construct different hypergraph Laplacian matrices, and then develop a suite of clustering methods that use these incidence matrices and Laplacians for hypergraph clustering. Using 20Newsgroup, U.S. patent, Reuters' Corpus Volume 1, and genetics data sets, we compare the performance of these clustering algorithms experimentally against a variety of existing hypergraph clustering methods. We show that the proposed methods produce higher-quality clusters.
Quantum Random Walk Simulator Using Ultrafast Optical Switches
Quantum random walk processes have many intriguing applications in high energy physics including the simulation of parton shower evolution. We will present the design and initial results of a fiber loop time-bin quantum walk architecture using the hardware platform already in operation at the Fermilab Quantum Network in which the state of the photon is defined by its time-of-arrival. The fiber loop consists of an unbalanced Mach-Zehnder interferometer implemented using an ultrafast electro-optical switch. The input switch controls the photon path within the interferometer, while the output switch will direct the photon back into the interferometer or to single photon detectors to measure the probability distribution of arrival times. Depending on which path the photon takes each pass through the loop, its wave function will interfere on these optical switches similar to quantum interference on a beam splitter. This work is an important step towards utilizing real-world advantages of quantum information protocols to solve problems in high energy physics.
A Finite Difference informed Random Walk solver for simulating radiation defect evolution in polycrystalline structures with strongly inhomogeneous diffusivity
Diffusivity of species and defects on grain boundaries is usually several orders of magnitude larger than that inside grains. Such strongly inhomogeneous diffusivity requires prohibitively high computational demands for modeling microstructural evolution. Here, this paper presents a highly-efficient numerical solver, combining the Finite Difference method and Random Walk model, designed for accurately modeling strongly inhomogeneous diffusion within polycrystalline structures. The proposed solver, termed Finite Difference informed Random Walk (FDiRW), integrates a customized Finite Difference (cFD) scheme tailored for fast diffusion along thin grain boundaries represented by a single-layer of nodes. Numerical experiments demonstrate that the FDiRW solver achieves an impressive efficiency gain of 1560x compared to traditional Finite Difference methods while maintaining accuracy, making it feasible for personal computer machines to handle diffusional systems with strongly inhomogeneous diffusivity across static polycrystalline microstructures. The model has been successfully applied to simulate radiation defect evolution, showcasing its scalability to engineering scales in both length and time dimensions.
Neuromorphic scaling advantages for energy-efficient random walk computations
Computing stands to be radically improved by neuromorphic computing (NMC) approaches inspired by the brain's incredible efficiency and capabilities. Most NMC research, which aims to replicate the brain's computational structure and architecture in man-made hardware, has focused on artificial intelligence; however, less explored is whether this brain-inspired hardware can provide value beyond cognitive tasks. We demonstrate that high-degree parallelism and configurability of spiking neuromorphic architectures makes them well-suited to implement random walks via discrete time Markov chains. Such random walks are useful in Monte Carlo methods, which represent a fundamental computational tool for solving a wide range of numerical computing tasks. Additionally, we show how the mathematical basis for a probabilistic solution involving a class of stochastic differential equations can leverage those simulations to provide solutions for a range of broadly applicable computational tasks. Despite being in an early development stage, we find that NMC platforms, at a sufficient scale, can drastically reduce the energy demands of high-performance computing platforms.
Oscillations and random walk of the soliton core in a fuzzy dark matter halo
We report that a fuzzy dark matter halo consists of a soliton core close to the center and a Navarro-Frenk-White–like density profile in the outer region. Previous investigations found that the soliton core exhibits temporal oscillations and random walk excursions around the halo center. Analyzing a set of numerical simulations, we show that both phenomena can be understood as the results of wave interference - a suitable superposition of the ground (solitonic) state and excited states in a fixed potential suffices to account for the main features of these phenomena. Such an eigenmode analysis can shed light on the evolution of a satellite halo undergoing tidal disruption. As the outer halo is stripped away, reducing the amplitudes of the excited states, the ground state evolves adiabatically. This suggests diminished soliton oscillations and random walk excursions, an effect to consider in deducing constraints from stellar heating.
Correction: The nanocaterpillar's random walk: diffusion with ligand–receptor contacts
Correction for ‘The nanocaterpillar's random walk: diffusion with ligand–receptor contacts’ by Sophie Marbach et al. , Soft Matter , 2022, 18 , 3130–3146, DOI: https://doi.org/10.1039/D1SM01544C.
Aging power spectrum of membrane protein transport and other subordinated random walks
Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion processes in complex systems are usually not stationary, the traditional Wiener-Khinchin theorem for the analysis of power spectral densities is invalid. Here, we employ a recently developed tool named aging Wiener-Khinchin theorem to derive the power spectral density of fractional Brownian motion coexisting with a scale-free continuous time random walk, the two most typical anomalous diffusion processes. Using this analysis, we characterize the motion of voltage-gated sodium channels on the surface of hippocampal neurons. Our results show aging where the power spectral density can either increase or decrease with observation time depending on the specific parameters of both underlying processes.
Two Problems in Applied Stochastic Processes: Scaling Limits of the Multiple Range in 1D Random Walk and Uncertainty Quantification in Satellite Imagery
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"Spectrally gapped" random walks on networks: a Mean First Passage Time formula
We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between a source and a target node of a directed and weighted network. The formula does not require any matrix inversion, and it takes as only input the transition probabilities into the target node. It is derived from the calculation of the average resolvent of a deformed ensemble of random sub-stochastic matrices H=\langle H\rangle +\delta H H = ⟨ H ⟩ + δ H , with \langle H\rangle ⟨ H ⟩ rank- 1 1 and non-negative. The accuracy of the formula depends on the spectral gap of the reduced transition matrix, and it is tested numerically on several instances of (weighted) networks away from the high sparsity regime, with an excellent agreement.