scholarly journals Small Subgraphs in the Trace of a Random Walk

10.37236/6169 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Peleg Michaeli
Keyword(s):  
Random Walk ◽  
Random Graph ◽  
Complete Graph ◽  
Combinatorial Properties ◽  
Base Graph

We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.

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.

Some remarks about extreme degrees in a random graph

1986 ◽  
Vol 100 (1) ◽  
pp. 167-174 ◽  
Author(s):  
Zbigniew Palka
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Probability Distributions ◽  
Supplementary Information ◽  
Random Subgraph

Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

On a Conjecture of Bollobás and Brightwell Concerning Random Walks on Product Graphs

1998 ◽  
Vol 7 (4) ◽  
pp. 397-401 ◽  
Author(s):  
OLLE HÄGGSTRÖM
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Complete Graph ◽  
Continuous Time ◽  
Product Graph ◽  
Product Graphs ◽  
Continuous Time Random Walks ◽  
Discrete Time Case

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

ON THE COMBINATORICS OF BINARY SERIES-PARALLEL GRAPHS

2016 ◽  
Vol 30 (2) ◽  
pp. 244-260
Author(s):  
Micha Hofri ◽  
Chao Li ◽  
Hosam Mahmoud
Keyword(s):  
Random Walk ◽  
Structural Equation ◽  
Regular Expression ◽  
Average Length ◽  
Probability Model ◽  
Order Number ◽  
Systematic Method ◽  
Large Graphs ◽  
Combinatorial Properties ◽  
Transportation Modeling

Binary series-parallel (BSP) graphs have applications in transportation modeling, and exhibit interesting combinatorial properties. This work is limited to the second aspect. The BSP graphs of a given size – measured in edges – are enumerated. Under a uniform probability model, we investigate the asymptotic distribution of the order (number of vertices) and the asymptotic average length of a random walk (under different strategies) of large graphs of the same size. The systematic method throughout is to define the graphs, and the features we evaluate by a structural equation (equivalent to a regular expression). The structural equation is translated into an equation for a generating function, which is then analyzed.

scholarly journals Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

2013 ◽  
Vol 50 (4) ◽  
pp. 1117-1130
Author(s):  
Stephen Connor
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Complete Graph ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Coupling Inequality ◽  
Maximal Coupling

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

scholarly journals A train track directed random walk on Out(Fr)

2015 ◽  
Vol 25 (05) ◽  
pp. 745-798 ◽  
Author(s):  
Ilya Kapovich ◽  
Catherine Pfaff
Keyword(s):  
Random Walk ◽  
Complete Graph ◽  
Outer Space ◽  
Simple Random Walk ◽  
Index Formula ◽  
Positive Probability ◽  
Train Track ◽  
The Ideal

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element φn ∈ Out (Fr), obtained after n steps of a simple random walk on Out (Fr), is fully irreducible with probability tending to 1 as n → ∞. In this paper, we construct a natural "train track directed" random walk 𝒲 on Out (Fr) (where r ≥ 3). We show that, for the element φn ∈ Out (Fr), obtained after n steps of this random walk, with asymptotically positive probability the element φn has the following properties: φn is an ageometric fully irreducible, which admits a train track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that φn has "rotationless index" [Formula: see text] (so that the geometric index of the attracting tree Tφn of φn is 2r - 3), has index list [Formula: see text] and the ideal Whitehead graph being the complete graph on 2r - 1 vertices, and that the axis bundle of φn in the Outer space CV r consists of a single axis.

scholarly journals Ramsey Games Against a One-Armed Bandit

2003 ◽  
Vol 12 (5-6) ◽  
pp. 515-545 ◽  
Author(s):  
Ehud Friedgut ◽  
Yoshiharu Kohayakawa ◽  
Vojtěch Rodl ◽  
Andrzej Rucinski ◽  
Prasad Tetali
Keyword(s):  
Random Graph ◽  
Positive Constant ◽  
Complete Graph ◽  
Upper Bound ◽  
High Probability ◽  
Random Sequence ◽  
Extreme Case ◽  
Online Version ◽  
Expected Length ◽  
Person Game

We study the following one-person game against a random graph process: the Player's goal is to 2-colour a random sequence of edges of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.While it is not hard to prove that the expected length of this game is about , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with edges, where c is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round.

scholarly journals Exact two-point resistance, and the simple random walk on the complete graph minus N edges

2012 ◽  
Vol 327 (12) ◽  
pp. 3116-3129 ◽  
Author(s):  
Noureddine Chair
Keyword(s):  
Random Walk ◽  
Complete Graph ◽  
Simple Random Walk
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