The Hitting Times of Random Walks on Bicyclic Graphs

2021 ◽  
Author(s):  
Xiaomin Zhu ◽  
Xiao-Dong Zhang
Keyword(s):  
Random Walks ◽  
Hitting Times ◽  
Bicyclic Graphs

On a Result of Aleliunas et al. Concerning Random Walks on Graphs

1990 ◽  
Vol 4 (4) ◽  
pp. 489-492 ◽  
Author(s):  
José Luis Palacios
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Proof ◽  
Hitting Times ◽  
Initial State ◽  
Random Walks On Graphs ◽  
Minimum Number

Aleliunas et al. [3] proved that for a random walk on a connected raph G = (V, E) on N vertices, the expected minimum number of steps to visit all vertices is bounded by 2|E|(N - 1), regardless of the initial state. We give here a simple proof of that result through an equality involving hitting times of vertices that can be extended to an inequality for hitting times of edges, thus obtaining a bound for the expected minimum number of steps to visit all edges exactly once in each direction.

Expected hitting times for random walks on weak products of graphs

1999 ◽  
Vol 43 (1) ◽  
pp. 33-39 ◽  
Author(s):  
Bárbara González-Arévalo ◽  
José Luis Palacios
Keyword(s):  
Random Walks ◽  
Hitting Times

On overshoots and hitting times for random walks

1999 ◽  
Vol 36 (2) ◽  
pp. 593-600
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Hitting Time ◽  
Hitting Times ◽  
First Hitting Time ◽  
Fixed Integer ◽  
Ladder Heights ◽  
Regenerative Sets

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

scholarly journals Hitting Times of Walks on Graphs through Voltages

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
José Luis Palacios ◽  
Eduardo Gómez ◽  
Miguel Del Río
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Hitting Times ◽  
Reciprocity Principle ◽  
Triangular Inequality ◽  
Random Walks On Graphs ◽  
Necklace Graph

We derive formulas for the expected hitting times of general random walks on graphs, in terms of voltages, with very elementary electric means. Under this new light we revise bounds and hitting times for birth-and-death Markov chains and for walks on graphs with cutpoints, and give some exact computations on the necklace graph. We also prove Tetali’s formula for hitting times without making use of the reciprocity principle. In fact this principle follows as a corollary of our argument that also yields as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour.

Exponential martingales and Wald's formulas for two-queue networks

1986 ◽  
Vol 23 (3) ◽  
pp. 812-819 ◽  
Author(s):  
François Baccelli
Keyword(s):  
Random Walks ◽  
Practical Interest ◽  
Hitting Times ◽  
Functional Relations ◽  
Exponential Martingale ◽  
Number Of Customers

An exponential martingale is defined for a class of random walks in the positive quarter lattice which are associated with a wide variety of Markovian two-queue networks. Balance formulas generalizing Wald's exponential identity are derived from the regularity of several types of hitting times with respect to this martingale. In a queuing context, these formulas can be interpreted as functional relations of practical interest between the number of customers at certain epochs and the utilization of the queues up to these epochs.

Hitting Times for Random Walks on Sierpiński Graphs and Hierarchical Graphs

2020 ◽  
Vol 63 (9) ◽  
pp. 1385-1396
Author(s):  
Yi Qi ◽  
Yuze Dong ◽  
Zhongzhi Zhang ◽  
Zhang Zhang
Keyword(s):  
Random Walks ◽  
Local Area Networks ◽  
Structural Difference ◽  
Local Area ◽  
Hitting Times ◽  
Topological Properties ◽  
Sierpiński Graphs ◽  
Model Complex ◽  
Self Similar ◽  
Processing Architectures

Abstract The Sierpiński graphs and hierarchical graphs are two much studied self-similar networks, both of which are iteratively constructed and have the same number of vertices and edges at any iteration, but display entirely different topological properties. Both graphs have a large variety of applications: Sierpiński graphs have a close connection with WK-recursive networks that are employed extensively in the design and implementation of local area networks and parallel processing architectures, while hierarchical graphs can be used to model complex networks. In this paper, we study hitting times for several absorbing random walks in Sierpiński graphs and hierarchical graphs. For all considered random walks, we determine exact solutions to hitting times for both graphs. The obtained explicit expressions indicate that the hitting times in both graphs behave quite differently. We show that the structural difference of the graphs is responsible for the disparate behaviors of their hitting times.

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