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.

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

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.

Calculations of first passage time of delayed tree-like networks

2015 ◽  
Vol 29 (28) ◽  
pp. 1550200
Author(s):  
Shuai Wang ◽  
Weigang Sun ◽  
Song Zheng
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Passage Time ◽  
The Novel ◽  
Mean First Passage Time ◽  
First Passage ◽  
Initial Node ◽  
Network Parameters ◽  
Self Similar ◽  
The Mean

In this paper, we study random walks in a family of delayed tree-like networks controlled by two network parameters, where an immobile trap is located at the initial node. The novel feature of this family of networks is that the existing nodes have a time delay to give birth to new nodes. By the self-similar network structure, we obtain exact solutions of three types of first passage time (FPT) measuring the efficiency of random walks, which includes the mean receiving time (MRT), mean sending time (MST) and mean first passage time (MFPT). The obtained results show that the MRT, MST and MFPT increase with the network parameters. We further show that the values of MRT, MST and MFPT are much shorter than the nondelayed counterpart, implying that the efficiency of random walks in delayed trees is much higher.

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.

Scalings of first-return time for random walks on generalized and weighted transfractal networks

2019 ◽  
Vol 33 (26) ◽  
pp. 1950306
Author(s):  
Qin Liu ◽  
Weigang Sun ◽  
Suyu Liu
Keyword(s):  
Random Walks ◽  
Large Scale ◽  
Probability Generating Function ◽  
Return Time ◽  
Network Size ◽  
Large Network ◽  
Large Scale Network ◽  
Scale Network ◽  
The Mean ◽  
First Return Time

The first-return time (FRT) is an effective measurement of random walks. Presently, it has attracted considerable attention with a focus on its scalings with regard to network size. In this paper, we propose a family of generalized and weighted transfractal networks and obtain the scalings of the FRT for a prescribed initial hub node. By employing the self-similarity of our networks, we calculate the first and second moments of FRT by the probability generating function and obtain the scalings of the mean and variance of FRT with regard to network size. For a large network, the mean FRT scales with the network size at the sublinear rate. Further, the efficiency of random walks relates strongly with the weight factor. The smaller the weight, the better the efficiency bears. Finally, we show that the variance of FRT decreases with more number of initial nodes, implying that our method is more effective for large-scale network size and the estimation of the mean FRT is more reliable.

scholarly journals Cutoff phenomena for random walks on random regular graphs

2010 ◽  
Vol 153 (3) ◽  
pp. 475-510 ◽  
Author(s):  
Eyal Lubetzky ◽  
Allan Sly
Keyword(s):  
Random Walks ◽  
Regular Graphs
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