Random walks on networks: Cumulative distribution of cover time

This paper describes an investigation of analytical formulas for parameters in random walks. Random walks are used to model situations in which an object moves in a sequence of steps in randomly chosen directions. Given a graph and a starting point, we select a neighbor of it at random, and move to this neighbor; then we select a neighbor of this point at random, and move to it etc. It is a fundamental dynamic process that arise in many models in mathematics, physics, informatics and can be used to model random processes inherent to many important applications. Different aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of hitting time, commute time, cover time are discussed in various works. In some papers, authors have derived an analytical expression for the distribution of the cover time for a random walk over an arbitrary graph that was tested for small values of n. However, this work will show the simplified analytical expressions for distribution of hitting time, commute time, cover time for bigger values of n. Moreover, this work will present the probability mass function and the cumulative distribution function for hitting time, commute time.

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A tight upper bound on the cover time for random walks on graphs

Random Structures and Algorithms ◽

10.1002/rsa.3240060106 ◽

1995 ◽

Vol 6 (1) ◽

pp. 51-54 ◽

Cited By ~ 82

Author(s):

Uriel Feige

Keyword(s):

Random Walks ◽

Upper Bound ◽

Cover Time ◽

Random Walks On Graphs

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Series expansions for the all-time maximum of α-stable random walks

Advances in Applied Probability ◽

10.1017/apr.2016.26 ◽

2016 ◽

Vol 48 (3) ◽

pp. 744-767

Author(s):

Clifford Hurvich ◽

Josh Reed

Keyword(s):

Random Walk ◽

Random Walks ◽

Queueing Theory ◽

Cumulative Distribution Function ◽

Random Variable ◽

Cumulative Distribution ◽

Series Expansions ◽

Stable Random Variables ◽

The Mean ◽

The Stability

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

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A Generalized Cover Time for Random Walks on Graphs

Formal Power Series and Algebraic Combinatorics ◽

10.1007/978-3-662-04166-6_10 ◽

2000 ◽

pp. 113-124 ◽

Cited By ~ 3

Author(s):

Cyril Banderier ◽

Robert P. Dobrow

Keyword(s):

Random Walks ◽

Cover Time ◽

Random Walks On Graphs

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

Advances in Applied Probability ◽

10.1017/s0001867800009836 ◽

2000 ◽

Vol 32 (01) ◽

pp. 177-192 ◽

Cited By ~ 3

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.

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

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/ac3a34 ◽

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.

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