Minimization of spatial cover times for impaired self-avoiding random walks: the mirage effect

2021 ◽  
Vol 2021 (6) ◽  
pp. 063404
Author(s):  
Daniel Campos ◽  
Javier Cristín ◽  
VicenÇ Méndez
Keyword(s):  
Random Walks ◽  
Mirage Effect ◽  
Cover Times

The ruin problem and cover times of asymmetric random walks and Brownian motions

2000 ◽  
Vol 32 (01) ◽  
pp. 177-192 ◽  
Author(s):  
K. S. Chong ◽  
Richard Cowan ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Cover Time ◽  
Brownian Motions ◽  
Brownian Motion With Drift ◽  
And Brownian Motion ◽  
Cover Times ◽  
Ruin Problem

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

scholarly journals Analytical results for the distribution of cover times of random walks on random regular graphs

2021 ◽  
Author(s):  
Ido Tishby ◽  
Ofer Biham ◽  
Eytan Katzav
Keyword(s):  
Random Walks ◽  
Cumulative Distribution ◽  
Large Network ◽  
Regular Graphs ◽  
Time Step ◽  
Single Node ◽  
Initial Node ◽  
Analytical Results ◽  
Partial Cover ◽  
Cover Times

Abstract We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of N nodes of degree c (c ≥ 3). Starting from a random initial node at time t = 1, at each time step t ≥ 2 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time TCis the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution Pt(S = s) of the number of distinct nodes s visited by an RW up to time t and solve it analytically. Inserting s = N we obtain the cumulative distribution of cover times, namely the probability P (TC ≤ t) = Pt(S = N) that up to time t an RW will visit all the N nodes in the network. Taking the large network limit, we show that P (TC ≤ t) converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times P (TPC,k = t), which is the probability that at time t an RW will complete visiting k distinct nodes. We also calculate the distribution of random cover (RC) times P (TRC,k = t), which is the probability that at time t an RW will complete visiting all the nodes in a subgraph of k randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.

scholarly journals Cover times for Brownian motion and random walks in two dimensions

2004 ◽  
Vol 160 (2) ◽  
pp. 433-464 ◽  
Author(s):  
Amir Dembo ◽  
Yuval Peres ◽  
Jay Rosen ◽  
Ofer Zeitouni
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Two Dimensions ◽  
Cover Times

Extremal cover times for random walks on trees

1990 ◽  
Vol 14 (5) ◽  
pp. 547-554 ◽  
Author(s):  
Graham Brightwell ◽  
Peter Winkler
Keyword(s):  
Random Walks ◽  
Cover Times

scholarly journals The power of two choices for random walks

2021 ◽  
pp. 1-28
Author(s):  
Agelos Georgakopoulos ◽  
John Haslegrave ◽  
Thomas Sauerwald ◽  
John Sylvester
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Graph ◽  
Optimal Strategy ◽  
Cover Time ◽  
Bounded Degree ◽  
Graph Classes ◽  
Algorithmic Question ◽  
Cover Times ◽  
Bounded Degree Trees

Abstract We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

scholarly journals The hitting and cover times of random walks on finite graphs using local degree information

2009 ◽  
Vol 410 (1) ◽  
pp. 94-100 ◽  
Author(s):  
Satoshi Ikeda ◽  
Izumi Kubo ◽  
Masafumi Yamashita
Keyword(s):  
Random Walks ◽  
Finite Graphs ◽  
Local Degree ◽  
Cover Times

scholarly journals The cover times of random walks on random uniform hypergraphs

2013 ◽  
Vol 509 ◽  
pp. 51-69 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Tomasz Radzik
Keyword(s):  
Random Walks ◽  
Uniform Hypergraphs ◽  
Cover Times
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