# Random Walks With Memory

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### Pierre Tarr es Curriculum vitˆ - Research Research NYU

of a Workshop on Random walks with memory at CIRM, Luminy, May 2017 Organizer of a parallel session at the Conference Stochastic Processes and Applications , Oxford, July 2015

### DrunkardMob: Billions of Random Walks on Just a PC

Random walks and recommender systems: Mod-els based on random walks on a graph are popular in the recommender system research due to their scalability. This work was initially motivated by the problem of recommend-ing friends or connections in a social network, called the link prediction problem [23]. Perhaps the most common ap-

### GPU-Friendly Floating Random Walk Algorithm for Capacitance

integration described with floating random walks. The FRW algorithm has the advantages of lower memory usage, more scalability for large structures and tunable accuracy, compared with the deterministic methods. To date, the FRW algorithm has evolved to several commercial capacitance solvers (e.g. the QuickCapTM). Recently, the FRW algorithm

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### Random Walk Problems Motivated by Statistical Physics

equlibirum statistical physics: self-avoiding random walk, loop-erased random walk, and intersections of paths of simple random walks. 1. Introduction A rich source of challenging problems in probability has been statistical physics. In this article, I will consider three related models of random walks with self-

### Joshua T. Abbott Joseph L. Austerweil

data. Random walks provide an alternative account of how people search their memory, postulating an undirected rather than a strategic search process. We show that results resembling optimal foraging are produced by random walks when related items are close together in the semantic network.

### B RANDOM WALKS

1.2.Branching random walks These notes are devoted to the (discrete-time, one-dimensional) branching random walk, which is a natural extension of the Galton Watson process in the spatial sense. The distribution of the branching random walk is governed by a random N-tuple Ξ := (ξi,1 ≤ i≤ N) of real numbers, where Nis also random and can be

### Memory Effects in Brownian Motion, Random Walks under

Memory E↵ects in Brownian Motion, Random Walks under Conﬁning Potentials, and Relaxation of Quantum Systems by Matthew Chase B.A, Colorado College, Colorado Springs, CO, May, 2009 M.S., Physics, University of New Mexico, Albuquerque, NM, 2014 DISSERTATION Submitted in Partial Fulﬁllment of the Requirements for the Degree of Doctor of

### Asymptotic behaviour of random walks with long memory

long memory of the random walk or diﬀusion is induced by a self-interaction mechanism deﬁned locally in a natural way in terms of the local time (or occupation time) process. The asymptotic scaling behaviour of self-interacting random walks and processes has

### Maximum likelihood estimators and random walks in long memory

cesses with long memory and we construct maximum likelihood estimators (MLE) for the drift parameter. Our approach is based in the non-Gaussian case on the approximation by random walks of the driving noise. We study the asymptotic behavior of the estimators and we give some numerical simulations to illustrate our results.

### PowerWalk: Scalable Personalized PageRank via Random Walks

the simulation of random walks is very time-consuming. Also, the precomputed database is usually too large to ﬁt in the main memory for large graphs which makes this method undesirable in practice. Bahmani et al. [5] suggested to concatenate the short random walks in the precomputed database to answer online queries. This exten-

### Analysis of Random Walks using Tabu Lists

2.1 Tabu Random Walks A tabu random walk on a simple graph is a partially self-avoiding random walk, where the walker is endowed with a ﬁnite memory and can jump from a node to another, pro-vided that they are neighbors. The memory of the walker, called tabu list, contains a part of the vertices already visited by the walker.

### Sudden onset of log-periodicity and superdiffusion in non

random walks are characterized by complete memory of the entire history. Subsequently, some of us modiﬁed the model with the intent of studying the eﬀects of loss of memory of the distant past [5] and of the recent past [6,7]. The latter model has been analytically studied by Kenkre [8], who coined the term Alzheimer walk , taking note of

### IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED

computational time and the bottleneck of memory usage. The FRW algorithm for capacitance extraction, presented as a 2-D version, was proposed in 1992 [9]. Its basic idea is to convert the calculation of conductor charge to the Monte Carlo (MC) integration performed with ﬂoating random walks. In the

### Examining Search Processes in Low and High Creative

of creativity using random walks. We formalize the search process proposed by the associative theory as an uncontrolled random walk, and predict that (on average) a random walk over the semantic network of HSC individuals will visit more nodes that are weaker in similarity than an equivalent length random walk over the semantic network

### Physics Reports Random walks and diffusion on networks

Random walks have been studied for many decades on both regular lattices and (especially in the last couple of decades) on networks with a variety of structures. In the present article, we survey the theory and applications of random walks on networks, restricting ourselves to simple cases of single and non-adaptive random walkers.

### GraphWalker: An I/O-Efficient and Resource-Friendly Graph

ØPerformance of random walks with different number of walks Fix walk length as 10 GraphWalkerachieves 16x-70x speedup. GraphWalkeris also capable to support huge graphs and massive walks. GraphWalkerfinishes running 1010walks on the largest dataset CrawlWebwithin around one hour.

### Memory in network flows and its effects on spreading dynamics

Memory in network ﬂows and its effects on spreading dynamics and community detection Martin Rosvall1, Alcides V. Esquivel1, Andrea Lancichinetti1,2, Jevin D. West1,3 & Renaud Lambiotte4 Random walks on networks is the standard tool for modelling spreading processes in social and biological systems.

### Prediction with a Short Memory

networks, such as Long Short-Term Memory (LSTM) networks [2, 3]. Other recently popular models that have explicit notions of memory include neural Turing machines [4], memory networks [5], differentiable neural computers [6], attention-based models [7, 8], etc. These models have been quite successful (see e.g. [9, 10]); never-

### Random Walks and Electric Resistance on Distance-Regular Graphs

Connection with random walks Let g(z) = Pz(v1beforev0) denote the probability that a random walk, started at z, strikes v1 before hitting v0. Random walk has no memory, so Pz(v1 before v0) = 1 d Px 1 (z1 before z0)+:::+ 1 d Px d (z1 before z0) (4) where x1;:::;xd are the points adjacent to z. This is the same

### RandomWalk.ppt - UIUC

Random Walks Example from A&T 110-123 Markov chains, detailed balanceand transition rules. Consider two states Sm and Sn linked by a transition probability P(m n). Suppose the reliability of your computer follows a certain pattern. If it is up and running on day, then it has a 60% chance of running on the next. If, however, it is

### Curriculum Vitae SWEE HONG CHAN - math.ucla.edu

2018 Nov.Duke Probability Seminar, Durham, USA, Random walks with local memory in the square lattice 2018 Sep.Penn/Temple Probability Seminar, Philadelphia, USA, In between random walk and rotor walk in the square lattice 2018 JuneFifth IMS Asia Paci c Rim Meeting (IMS-APRM), Singapore, Singapore, In between random

### University of Pennsylvania

Chapter 8. Non-Stationarity: Integration, Cointegration and Long Memory 126 8.1 Random Walks as the I(1) Building Block: The Beveridge-Nelson Decomposition126 8.2 Stochastic vs. Deterministic Trend127 8.3 Unit Root Distributions128 8.4 Univariate and Multivariate Augmented Dickey-Fuller Representations130 8.5 Spurious Regression131

### Random Walks with Memory Applied to Grand Slam Tennis Matches

Random Walks with Memory Applied to Grand Slam Tennis Matches Modeling Tom´ a sKou rim Institute of Information Theory and Automation, Czech Academy of Sciences Prague, Czech Republic [email protected] Abstract The contribution presents a model of a random walk with varying transition probabil-

### Investigating Mixed Memory-Reinforcement Models for Random Walks

Random Walks in Biology De nition A random walk is a path that consists of a series of random steps. Examples Path of a molecule in a gas Motion of a slime mold towards food Movement of ants between food source and anthill Not necessarily purely random Ria Das, Phillips Exeter Academy Mentor: Andrew Rzeznik Investigating Mixed Memory

### Random Walks in Peer-to-Peer Networks

Random Walks in Peer-to-Peer Networks Christos Gkantsidis, Milena Mihail, and Amin Saberi College of Computing Georgia Institute of Technology Atlanta, GA Email: gantsich, mihail, saberi @cc.gatech.edu AbstractŠWe quantify the effectiveness of random walks for searching and construction of unstructured peer-to-peer (P2P) networks. For

### Multi-Dimensional Elephant Random Walk with Coupled Memory

referred to as Elephant Random Walks, ERW, was pro-posed and analytically solved by Schutz and Trimper in 2004 [7]. In this model, a one-dimensional elephant takes steps to the right or to the left and the probability of each step depends on the whole history of the elephant. The memory e ect is due to a single parameter p2[0;1]

### Vertex-reinforced Random Walk for Network Embedding

neighbors and the memory. This random walk has a strong mathematical foundation based on the vertex-reinforced random walk (VRRW) [22]. Di erent from the original VRRW, our random walk is guided by two factors. The rst one is to guide the random walk to follow the memory, which is the same as VRRW. However, methods that only based on memory

### Random Walks - Sharif

Memory Loss in Random Walks Theorem. If we run the walk for sufficiently many steps, the probability of finding it at any given vertex will converge to a fixed value regardless of where the walk started. Let s generalize the random walk. G is directed graph. For each vertex u of G we have a probability P uv of

### Random walks

Random walks Random walks are one of the basic objects studied in probability theory. The moti-vation comes from observations of various random motions in physical and biolog-ical sciences. The most well-known example is the erratic motion of pollen grains immersed in a ﬂuid observed by botanist Robert Brown in 1827 caused, as

### J Mathematical Methods in Engineering AUCTORES Journal of

exponential memory profile random walk model (Alves et al., 2014), the exponential memory profile random walk model ( Moura et al., 2016) and the random walk model with binomial memory profile (Diniz et al., 2017). that the st In this class of random walks memory is an important feature. The memory is formed by a set of random variables

### Massively Parallel Algorithms for Personalized PageRank

optimization techniques to avoid exploding the memory capacity by hierarchical sampling on large vertices and save sampling cost by pre-storing short random walks on vertices with a small de-gree. However, Monte-Carlo based methods [6, 24] incur a large number of random walks and can hardly measure the workload

### Wander Join: Online Aggregation via Random Walks

approach, in both internal and external memory, is based on ripple join, which is still very expensive and even needs unrealistic assumptions (e.g., tuples in a table are stored in random order). This paper proposes a new approach, the wander join algorithm, to the online aggregation problem by performing random walks over the underlying join

### Fast Query Execution for Retrieval Models Based on Path

sures are based on random walks on graphs, for instance lazy random walks [19] or personalized PageRank [6, 8] or Random Walk with Restart [25]. However, regular graph-walk based similarity measure is naive in the sense that the random walker does not distinguish the importance of diﬀer-ent paths. Relational data is usually annotated with a rich

### Random Walks with Memory Applied to Grand Slam Tennis Matches

Random Walks with Memory Applied to Grand Slam ennisT Matches Modeling Ing. omáT² Kou°im Institute of Information Theory and Automation, AS CR Prague 2.7.2019 Ing. omTá² Kou°im UTIA Random Walks with Memory Applied to Grand Slam ennisT Matches Modeling

### Non-Gaussian Distributions to Random Walk in the Context of

memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and

### random walks - Pennsylvania State University

cesses belong to a general class of random walks on lattices with weighted bonds or sites. Well-studied models include the self-avoiding random walk [15], the reinforced ran-dom walk [16], and the excited random walk [17]. These random walks with memory show unusual behavior, like anomalous di usion or spatial con nement.

### COUPLING TIMES FOR RANDOM WALKS WITH INTERNAL STATES

ABSTRACT.Using coupling techniques, we bound the mixing times of various walks with internal states. These random walks include the L-lattice, the Manhattan lattice, self-avoiding walks of ﬁnite memory, and walks on hexagonal lattices. We also investigate ideal couplings for arbitrary Markov chains of memory-2 over f0;1g.

### GraphWalker: An I/O-Efficient and Resource-Friendly Graph

dom walk features. To efﬁciently support parallel random walks, DrunkardMob [23]proposes several optimizations to reduce the memory usage of walk indexes so as to support a large amount of random walks. However, its scalability is still limited, e.g., it costs 2.3 hours to run one billion random walks with ten-step long on a medium-scale graph

### Pseudo Memory Eﬀects, Majorization and Entropy in Quantum

A quantum random walk on the integers exhibits pseudo memory eﬀects, in that its probability distribution after N steps is determined by reshuﬄing the ﬁrst N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy in-crease can be regarded as a consequence of the majorization