scholarly journals The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree

10.37236/8327 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Tony Johansson
Keyword(s):  
Random Walk ◽  
Regular Graph ◽  
Vertex Cover ◽  
Random Walk Process ◽  
Cover Time ◽  
Edge Cover ◽  
Biased Random Walk ◽  
Random Regular Graph ◽  
Odd Degree ◽  
Leading Term

We consider a random walk process on graphs introduced by Orenshtein and Shinkar (2014). At any time, the random walk moves from its current position along a previously unvisited edge chosen uniformly at random, if such an edge exists. Otherwise, it walks along a previously visited edge chosen uniformly at random. For the random $r$-regular graph, with $r$ a constant odd integer, we show that this random walk process has asymptotic vertex and edge cover times $\frac{1}{r-2}n\log n$ and $\frac{r}{2(r-2)}n\log n$, respectively, generalizing a result of Cooper, Frieze and the author (2018) from $r = 3$ to any odd $r\geqslant 3$. The leading term of the asymptotic vertex cover time is now known for all fixed $r\geqslant 3$, with Berenbrink, Cooper and Friedetzky (2015) having shown that $G_r$ has vertex cover time asymptotic to $\frac{rn}{2}$ when $r\geqslant 4$ is even.

scholarly journals Greedy Random Walk

2013 ◽  
Vol 23 (2) ◽  
pp. 269-289 ◽  
Author(s):  
TAL ORENSHTEIN ◽  
IGOR SHINKAR
Keyword(s):  
Random Walk ◽  
Random Process ◽  
Discrete Time ◽  
Complete Graph ◽  
Cover Time ◽  
Edge Cover ◽  
Adjacent Edge ◽  
Natural Families

We study a discrete time self-interacting random process on graphs, which we call greedy random walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not yet been crossed by the walker. At each step, the walker, being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walker jumps along it to the neighbouring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in$\mathbb{Z}^d$for alld≥ 3.

scholarly journals The Cover Time of Deterministic Random Walks

10.37236/439 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Tobias Friedrich ◽  
Thomas Sauerwald
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Cover Time ◽  
Short Term ◽  
Edge Cover ◽  
Graph Classes ◽  
Fixed Order ◽  
Deterministic Random Walk

The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how quickly this "deterministic random walk" covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes. Depending on the topology, the deterministic random walk can be asymptotically faster, slower or equally fast as the classic random walk. We also examine the short term behavior of deterministic random walks, that is, the time to visit a fixed small number of vertices or edges.

One-step random-walk process of nanoparticles in cement-based materials

2021 ◽  
Vol 28 (6) ◽  
pp. 1679-1691
Author(s):  
Ali Bahari ◽  
Aref Sadeghi-Nik ◽  
Elena Cerro-Prada ◽  
Adel Sadeghi-Nik ◽  
Mandana Roodbari ◽  
...  
Keyword(s):  
Random Walk ◽  
Random Walk Process ◽  
Cement Based Materials ◽  
One Step

A Biased Random Walk with Symmetry

1992 ◽  
Vol 65 (5) ◽  
pp. 330
Author(s):  
J. N. Boyd ◽  
P. N. Raychowdhury
Keyword(s):  
Random Walk ◽  
Biased Random Walk

COMPUTATIONAL INVESTIGATION OF THE EFFECTS OF SPHERICAL FILLER MORPHOLOGY AND LOADING ON DIFFUSION TORTUOSITY AND RUBBER PERMEABILITY

2013 ◽  
Vol 86 (2) ◽  
pp. 175-189 ◽  
Author(s):  
T. C. Gruber ◽  
S. D. Crossley ◽  
A. P. Smith
Keyword(s):  
Random Walk ◽  
Rolling Resistance ◽  
Diffusion Path ◽  
Filler Loading ◽  
Pressure Losses ◽  
Biased Random Walk ◽  
Gas Molecule ◽  
Path Lengths ◽  
Path Tortuosity ◽  
Spherical Filler

ABSTRACT Inflation pressure loss in tires degrades performance, raises rolling resistance, and reduces fuel economy. The incorporation of solid fillers, such as carbon black, at relatively high loadings in tire innerliners helps minimize these pressure losses by reducing innerliner permeability due to increases in average gas molecule diffusion path lengths (tortuosity), as well as reductions in diffusion pathway density (capacity). The effects of filler morphology and loading on diffusion path tortuosity can be explored by modeling biased random-walk diffusion through impermeable sphere-filled matrices. Modeled diffusion rate was found to decrease with increased filler loading, reduced filler sphere sizes, increased random-walk step sizes, and the aggregation of filler spheres. Initial correlations with limited empirical permeability measurements are used to validate the model approach.

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