Analogue Of Random Walk Research Articles

Fully Decentralized Federated Learning-Based On-Board Mission for UAV Swarm System

To handle the data explosion in the era of Internet-of-things, it is of interest to investigate the decentralized network, with the aim at relaxing the burden at the central server along with preserving data privacy. In this work, we develop a fully decentralized federated learning (FL) framework with an inexact stochastic parallel random walk alternating direction method of multipliers (ISPW-ADMM). Performing more efficient communication and enhanced privacy preservation compared with the current state-of-the-art, the proposed ISPW-ADMM can be partially immune to the effect of time-varying dynamic network and stochastic data collection, while still in fast convergence. Benefiting from the stochastic gradients and biased first-order moment estimation, the proposed framework can be applied to any decentralized FL tasks over time-varying graphs. Thus, to demonstrate the practicability of such a framework in providing fast convergence, high communication efficiency, noise robustness for a specific on-board mission to some extent, we study the extreme learning machine-based FL model beamforming design in unmanned aerial vehicle communications, as verified by the numerical simulations.

ThunderRW

As random walk is a powerful tool in many graph processing, mining and learning applications, this paper proposes an efficient in-memory random walk engine named ThunderRW. Compared with existing parallel systems on improving the performance of a single graph operation, ThunderRW supports massive parallel random walks. The core design of ThunderRW is motivated by our profiling results: common RW algorithms have as high as 73.1% CPU pipeline slots stalled due to irregular memory access, which suffers significantly more memory stalls than the conventional graph workloads such as BFS and SSSP. To improve the memory efficiency, we first design a generic step-centric programming model named Gather-Move-Update to abstract different RW algorithms. Based on the programming model, we develop the step interleaving technique to hide memory access latency by switching the executions of different random walk queries. In our experiments, we use four representative RW algorithms including PPR, DeepWalk, Node2Vec and MetaPath to demonstrate the efficiency and programming flexibility of ThunderRW. Experimental results show that ThunderRW outperforms state-of-the-art approaches by an order of magnitude, and the step interleaving technique significantly reduces the CPU pipeline stall from 73.1% to 15.0%.

Optimizing Random Walk Based Statistical Estimation over Graphs via Bootstrapping

Random walk sampling is often used to conduct statistical estimation over graphs. This paper develops an algorithmic framework to reduce the mean square error of such statistical estimation. Our algorithmic framework is inspired by that the mean square error can be decomposed into a sum of the bias and variance of the estimator. More specifically, we apply the bootstrapping technique to design a bias reduction algorithm. Our bias reduction algorithm only utilizes a small number of valid sub-samples, which can reduce more bias of the estimator but may increase the variance of the estimator significantly. We use multiple parallel random walks to reduce this variance such that it can be reduced to arbitrarily small by deploying a sufficient number of random walks. We provide theoretical guarantees and computational complexity analysis of our proposed bias reduction algorithms. Also, the proposed bias reduction algorithm is generic and can be applied to optimize a large class of random walk sampling algorithms. Extensive experiments on four public datasets confirm the effectiveness and computational efficiency of our proposed algorithmic framework under the mean square metric and beyond.

Decentralized Consensus Optimization Based on Parallel Random Walk

The alternating direction method of multipliers (ADMM) has recently been recognized as a promising approach for large-scale machine learning models. However, very few results study ADMM from the aspect of communication costs, especially jointly with running time. In this letter, we investigate the communication efficiency and running time of ADMM in solving the consensus optimization problem over decentralized networks. We first review the effort of random walk ADMM (W-ADMM), which reduces communication costs at the expense of running time. To accelerate the convergence speed of W-ADMM, we propose the parallel random walk ADMM (PW-ADMM) algorithm, where multiple random walks are active at the same time. Moreover, to further reduce the running time of PW-ADMM, the intelligent parallel random walk ADMM (IPW-ADMM) algorithm is proposed through integrating the Random Walk with Choice with PW-ADMM. By numerical results from simulations, we demonstrate that the proposed algorithms can be both communication efficient and fast in running speed compared with state-of-the-art methods.

Decentralized Consensus Optimization Based on Parallel Random Walk

The alternating direction method of multipliers (ADMM) has recently been recognized as a promising approach for large-scale machine learning models. However, very few results study ADMM from the aspect of communication costs, especially jointly with running time. In this letter, we investigate the communication efficiency and running time of ADMM in solving the consensus optimization problem over decentralized networks. We first review the effort of random walk ADMM (W-ADMM), which reduces communication costs at the expense of running time. To accelerate the convergence speed of W-ADMM, we propose the parallel random walk ADMM (PW-ADMM) algorithm, where multiple random walks are active at the same time. Moreover, to further reduce the running time of PW-ADMM, the intelligent parallel random walk ADMM (IPW-ADMM) algorithm is proposed through integrating the \textit{Random Walk with Choice} with PW-ADMM. By numerical results from simulations, we demonstrate that the proposed algorithms can be both communication efficient and fast in running speed compared with state-of-the-art methods.

Numerical modeling of flow and transport in the Bari industrial area by means of rough walled parallel plate and random walk models

Abstract. Modeling fluid flow and solute transport dynamics in fractured karst aquifers is one of the most challenging tasks in hydrogeology. The present study investigates the hotspots of groundwater contamination in the industrial area of Modugno (Bari – southern Italy), where the limestone aquifer has a fractured and karstic nature. A rough walled parallel plate model coupled with a geostatistical analysis to infer the values of the equivalent aperture has been implemented and calibrated on the basis of piezometric data. Using the random walk theory, the steady-state distribution of hypothetical contamination with the source at the hotspot has been carried out, reproducing a pollution scenario which is compatible with the observed one. From an analysis of the flow and transport pattern it is possible to infer that the anticline affecting the Calcare di Bari formation in direction ENE–WSW influences the direction of flow as well as the propagation of the contaminant. The results also show that the presence of nonlinear flow influences advection, in that it leads to a delay in solute transport with respect to the linear flow assumption. This is due to the non-constant distribution of solutes according to different pathways for fractured media which is related to the flow rate.

Cloud-Orchestrated Physical Topology Discovery of Large-Scale IoT Systems Using UAVs

Wireless sensor networks (WSNs) have been rapidly integrated into Internet of Things (IoT) systems, empowering rich and diverse applications such as large-scale environment monitoring. However, due to the random deployment of sensor nodes (SNs), physical topology of the WSNs cannot be controlled and typically remains unknown to the IoT cloud server. Therefore, in order to derive the physical topology at the cloud for effective real-time event detection, a cloud-orchestrated physical topology discovery scheme for large-scale IoT systems using unmanned aerial vehicles (UAVs) is proposed in this paper. More specifically, the large-scale monitoring area is first split into a number of subregions for UAV-enabled data collection. Within the subregions, parallel Metropolis-Hastings random walk (MHRW) is developed to gather the information of WSN nodes, including their IDs and neighbor tables. The collected information is then forwarded to the cloud through UAVs for the initial generation of logical topology. Thereafter, a network-wide 3-D localization algorithm is further developed based on the discovered logical topology and multidimensional scaling method (Topo-MDS), where the UAVs equipped with global positioning system are served as mobile anchors to locate the SNs. Simulation results indicate that the parallel MHRW improves both the efficiency and accuracy of logical topology discovery. In addition, the Topo-MDS algorithm dramatically improves the 3-D location accuracy, as compared to the existing algorithms in the literature.

The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. The behavior of the rotor-router is fully deterministic but its performance characteristics (cover time, return time) closely resemble the expected values of the corresponding parameters of the random walk. In this work we consider the setting in which multiple, indistinguishable agents are deployed in parallel in the nodes of the graph, and move around the graph in synchronous rounds, interacting with a single rotor-router system. We propose new techniques which allow us to perform a theoretical analysis of the multi-agent rotor-router model, and to compare it to the scenario of parallel independent random walks in a graph. Our main results concern the n-node ring, and suggest a strong similarity between the performance characteristics of this deterministic model and random walks. We show that on the ring the rotor-router with k agents admits a cover time of between varTheta (n^2 / k^2) in the best case and varTheta (n^2 / log k) in the worst case, depending on the initial locations of the agents, and that both these bounds are tight. The corresponding expected value of the cover time for k random walks, depending on the initial locations of the walkers, is proven to belong to a similar range, namely between varTheta (n^2 / (k^2/log ^2 k)) and varTheta (n^2 / log k). Finally, we study the limit behavior of the rotor-router system. We show that, once the rotor-router system has stabilized, all the nodes of the ring are always visited by some agent every varTheta (n / k) steps, regardless of how the system was initialized. This asymptotic bound corresponds to the expected time between successive visits to a node in the case of k random walks. All our results hold up to a polynomially large number of agents (1 le k < n^{1/11}).

Orbits of rotor-router operation and stationary distribution of random walks on directed graphs

The rotor-router model is a popular deterministic analogue of random walk. In this paper we prove that all orbits of the rotor-router operation have the same size on a strongly connected directed graph (digraph) and give a formula for the size. By using this formula we address the following open question about orbits of the rotor-router operation: Is there an infinite family of non-Eulerian strongly connected digraphs such that the rotor-router operation on each digraph has a single orbit?It turns out that on a strongly connected digraph the stationary distribution of the simple random walk coincides with the frequency of vertices in a rotor walk. In this common aspect a rotor walk simulates a random walk. This gives one similarity between two models on (finite) digraphs.

FrogWild!

We propose FrogWild, a novel algorithm for fast approximation of high PageRank vertices, geared towards reducing network costs of running traditional PageRank algorithms. Our algorithm can be seen as a quantized version of power iteration that performs multiple parallel random walks over a directed graph. One important innovation is that we introduce a modification to the GraphLab framework that only partially synchronizes mirror vertices. This partial synchronization vastly reduces the network traffic generated by traditional PageRank algorithms, thus greatly reducing the per-iteration cost of PageRank. On the other hand, this partial synchronization also creates dependencies between the random walks used to estimate PageRank. Our main theoretical innovation is the analysis of the correlations introduced by this partial synchronization process and a bound establishing that our approximation is close to the true PageRank vector. We implement our algorithm in GraphLab and compare it against the default PageRank implementation. We show that our algorithm is very fast, performing each iteration in less than one second on the Twitter graph and can be up to 7x faster compared to the standard GraphLab PageRank implementation.

Escape Rates for Rotor Walks in $\mathbb{Z}^d$

Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on ${\mathbb Z}^d$ for the initial configuration of all rotors aligned. If $n$ particles in turn perform rotor walks starting from the origin, we show that the number that escape (i.e., never return to the origin) is of order $n$ in dimensions $d \geq 3$ and of order $n/\log n$ in dimension $2$.

Parallel Random Walk for P/G Analysis

This paper introduced parallel computing techniques to improve random walk algorithm. The random walk problem was firstly explained by a formal model. And then, the parallel features of random walk algorithm were discussed in detail. A parallel random walk algorithm was proposed and applied to analyze the VLSI power grid. The time complexity and the main factors impacting on the execution time of algorithm were analyzed carefully. The experimental results proved that the parallel computing techniques could improve random walk algorithm effectively.

PERFORMANCE ANALYSIS AND EVALUATION OF RANDOM WALK ALGORITHMS ON WIRELESS NETWORKS

We propose a model of dynamically evolving random networks and give an analytical result of the cover time of the simple random walk algorithm on a dynamic random symmetric planar point graph. Our dynamic network model considers random node distribution and random node mobility. We analyze the cover time of the parallel random walk algorithm on a complete network and show by numerical data that k parallel random walks reduce the cover time by almost a factor of k. We present simulation results for four random walk algorithms on random asymmetric planar point graphs. These algorithms include the simple random walk algorithm, the intelligent random walk algorithm, the parallel random walk algorithm, and the parallel intelligent random walk algorithm. Our random network model considers random node distribution and random battery transmission power. Performance measures include normalized cover time, probability distribution of the length of random walks, and load distribution.

Analysis of Amino Acid Specific Energy Contributions to Native Conformations in High-Resolution Protein Structures

Because most proteins are marginally stable, each amino acid must contribute a near-optimal stabilization energy to the overall protein-structure. In this study we have analysed the specific energy contribution of each amino acid found in the native conformation of globular proteins. Using our free-energy forcefield PFF02 we devise a local energy measure on a per-amino acid basis and relax a population of 50 high resolution experimental globular protein structures on POEM@HOME with an evolutionary Monte Carlo energy minimization algorithm. The obtained set of relaxed structures is screened for locally stable segments, ruling out intrinsical disorder, and datasets of the energy ranges are assembled correlated with the position of the amino acid (interior, exterior, solvent-exposed).POEM (Protein Optimization using Energy Methods) identifies the native conformation of the protein as the global minimum of the protein free-energy forcefield PFF02, which stabilized the native conformation of all 32 monomeric proteins (without cofactors) against all decoys in the Rosetta decoy set. In addition we could fold a set of 13 proteins with helical, sheet and mixed secondary structure from completely unfolded conformations to near-native conformations, to an average 2.87 A resolution.The simulations we have conducted, were run on the POEM@HOME (http://boinc.fzk.de) volunteer computing architecture using a multiple population evolutionary strategy, which explores the free-energy surface in many parallel Monte-Carlo random walks. Various distinct temperature populations are evolved to the global free-energy minimum by balancing energy improvement and population diversity.

Performance of An All-Atom Free Energy Approach For Protein Structure Prediction

With the completion of sequencing efforts for many important genomes, protein structure and function prediction, emerge as important challenges. Physics-based or forcefield-based methods, which were initially believed to hold great promise for protein structure prediction, now play only a marginal role in the biannual comparative assessment of methods for protein structure prediction (CASP). Most of the models submitted to this computational experiment presently originate from similarity-based methods however structure prediction for low-homology targets leave significant room for improvement. Here we present some of the results from CASP9, were we combine physics based methods along with homology modeling. However, in case of low homology targets we generated libraries comprising of 10,000-20,000 decoys using the standard ROSETTA fragment assembly protocol and then energy-ranked using the POEM≅HOME, which explores the free-energy surface in many parallel Monte-Carlo random walks, the conformation which is near the native conformations from the large set of decoys was reliably selected.Some experimental structures of CASP9 targets have been released till date and the comparisons of our predictions are encouraging.

Rotor Walks on General Trees

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighboring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an infinite rooted tree, restarted from the root after each escape to infinity. We prove that the limiting proportion of escapes to infinity equals the escape probability for random walk, provided only finitely many rotors send the walker initially toward the root. For independently and identically distributed random initial rotor directions on a regular tree, the limiting proportion of escapes is either zero or the random walk escape probability, and undergoes a discontinuous phase transition between the two as the distribution is varied. In the critical case there are no escapes, but the walker's maximum distance from the root grows doubly exponentially with the number of visits to the root. We also prove that there exist trees of bounded degree for which the proportion of escapes eventually exceeds the escape probability by arbitrarily large $o(1)$ functions. No larger discrepancy is possible, while for regular trees, the discrepancy is at most logarithmic.

Estimates on the Speedup and Slowdown for a Diffusion in a Drifted Brownian Potential

We study a model of diffusion in a brownian potential. This model was firstly introduced by T. Brox (1986) as a continuous time analogue of random walk in random environment. We estimate the deviations of this process above or under its typical behavior. Our results rely on different tools such as a representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani's lemma, introduced at first by K. Kawazu and H. Tanaka (1997), and a decomposition of hitting times developed in a recent article by A. Fribergh, N. Gantert and S. Popov (2008). Our results are in agreement with their results in the discrete case.

Adaptive stochastic numerical scheme in parallel random walk models for transport problems in shallow water

This paper deals with the simulation of transport of pollutants in shallow water using random walk models and develops several computation techniques to speed up the numerical integration of the stochastic differential equations (SDEs). This is achieved by using both random time stepping and parallel processing. We start by considering a basic stochastic Euler scheme for integration of the diffusion and drift terms of the SDEs, with a strong order 1 in the strong sense. The errors due to this scheme depend on the location of the pollutant; it is dominated by the diffusion term near boundaries, and by the deterministic drift further away from the boundaries. Using a pair of integration schemes, one of strong order 1.5 near the boundary and one of strong order 2.0 elsewhere, we can estimate the error and approximate an optimal step size for a given error tolerance. The resulting algorithm is developed such that it allows for complete flexibility of the step size, while guaranteeing the correct Brownian behaviour. Modelling pollutants by non-interacting particles enables the use of parallel processing in the simulation. We take advantage of this by implementing the algorithm using the MPI library. The inherent asynchronic nature of the particle simulation, in addition to the parallel processing, makes it difficult to get a coherent picture of the results at any given points. However, by inserting internal synchronisation points in the temporal discretisation, the code allows pollution snapshots and particle counts to be made at times specified by the user.

All Atom Protein Folding Simulations With Distributed Computational Resources

On the basis of Anfinsen's thermodynamic hypothesis we have developed a free-energy forcefield PFF02 for all-atom de-novo protein simulation, which we recently implemented in a massively parallel computational environment. POEM (Protein Optimization using Energy Methods), which implements PFF02 along with several simulation protocols, identifies the native conformation of the protein as the global minimum of the protein free-energy forcefield. PFF02 was validated by stabilizing the native conformation of all 32 monomeric proteins in the Rosetta decoy sets (without cofactors) against the corresponding decoys. In addition we could fold a set of 24 proteins with helical, sheet and mixed secondary structure from completely unfolded conformations to near-native conformations to an average 2.87 Å resolution using traditional computational networks. The simulations we report here were performed on the [email protected] (http://boinc.fzk.de) volunteer computing architecture using a multiple population evolutionary strategy, which explores the free-energy surface in many parallel Monte-Carlo random walks. Various distinct temperature populations are evolved to the global free-energy minimum by balancing energy improvement and population diversity. This massively parallel algorithm enables the exploration of the free-energy landscape of relatively large proteins. Here we report simulations of three proteins that fold from completely extended conformations to native structural ensembles within experimental resolution. The largest of the three is the 43-amino acid fructose-repressor DNA-binding domain 1UXD. Analyzing the trajectory we find concomitant hydrophobic collapse and secondary structure formation. We observe intermediate collapsed states with native secondary structure content and few native tertiary contacts, which subsequently arrange into the native structure.

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work (Levine and Peres, Indiana Univ Math J 57(1):431–450, 2008). For the shape consisting of $n=\omega_d r^d$ sites, where ω d is the volume of the unit ball in $\mathbb{R}^d$ , we show that the inradius of the set of occupied sites is at least r − O(logr), while the outradius is at most r + O(r α ) for any α > 1 − 1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n = πr 2 particles, we show that the inradius is at least $r/\sqrt{3}$ , and the outradius is at most $(r+o(r))/\sqrt{2}$ . This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig.