scholarly journals Hypergroups and distance distributions of random walks on graphs

2021 ◽  
Vol 127 (1) ◽  
pp. 43-62
Author(s):  
Kenta Endo ◽  
Ippei Mimura ◽  
Yusuke Sawada
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Commutative Algebra ◽  
Transition Matrix ◽  
Operator Norm ◽  
Transition Matrices ◽  
Distance Distributions ◽  
Random Walks On Graphs ◽  
Matrix Products

Wildberger's construction enables us to obtain a hypergroup from a random walk on a special graph. We will give a probability theoretic interpretation to products on the hypergroup. The hypergroup can be identified with a commutative algebra whose basis is transition matrices. We will estimate the operator norm of such a transition matrix and clarify a relationship between their matrix products and random walks.

Determining Equivalent Resistance and Nodal Potentials of a Resistive Circuit Using a Reduced Transition Matrix

2020 ◽  
Vol 02 (01) ◽  
pp. 2050004
Author(s):  
Je-Young Choi
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Transition Matrix ◽  
Present Approach ◽  
Voltage Source ◽  
Equivalent Resistance ◽  
Electrical Circuits ◽  
Electric Potentials ◽  
Dominant Eigenvector

Several methods have been developed in order to solve electrical circuits consisting of resistors and an ideal voltage source. A correspondence with random walks avoids difficulties caused by choosing directions of currents and signs in potential differences. Starting from the random-walk method, we introduce a reduced transition matrix of the associated Markov chain whose dominant eigenvector alone determines the electric potentials at all nodes of the circuit and the equivalent resistance between the nodes connected to the terminals of the voltage source. Various means to find the eigenvector are developed from its definition. A few example circuits are solved in order to show the usefulness of the present approach.

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.

scholarly journals Random Walks in Hypergraph

2021 ◽  
Vol 15 ◽  
pp. 13-20
Author(s):  
Abdelghani Bellaachia ◽  
Mohammed Al-Dhelaan
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Transition Probabilities ◽  
Graph Representation ◽  
Real World Data ◽  
Data Set ◽  
Random Walks On Graphs ◽  
Text Ranking ◽  
Relationship Structure ◽  
The Relationship

Random walks on graphs have been extensively used for a variety of graph-based problems such as ranking vertices, predicting links, recommendations, and clustering. However, many complex problems mandate a high-order graph representation to accurately capture the relationship structure inherent in them. Hypergraphs are particularly useful for such models due to the density of information stored in their structure. In this paper, we propose a novel extension to defining random walks on hypergraphs. Our proposed approach combines the weights of destination vertices and hyperedges in a probabilistic manner to accurately capture transition probabilities. We study and analyze our generalized form of random walks suitable for the structure of hypergraphs. We show the effectiveness of our model by conducting a text ranking experiment on a real world data set with a 9% to 33% improvement in precision and a range of 7% to 50% improvement in Bpref over other random walk approaches.

scholarly journals Frogs and some other interacting random walks models

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Serguei Yu. Popov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Active Particle ◽  
Open Problems ◽  
Simple Version ◽  
Random Walks On Graphs ◽  
Interacting Random Walks ◽  
International Audience ◽  
Frog Model

International audience We review some recent results for a system of simple random walks on graphs, known as \emphfrog model. Also, we discuss several modifications of this model, and present a few open problems. A simple version of the frog model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active.

scholarly journals A random walk on the rook placements on a Ferrer's board

10.37236/1284 ◽  
1996 ◽  
Vol 3 (2) ◽  
Author(s):  
Phil Hanlon
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Rate Of Convergence ◽  
Spectral Methods ◽  
Transition Matrix ◽  
Rates Of Convergence ◽  
Sharp Estimates ◽  
Rook Placements ◽  
Ferrers Board

Let $B$ be a Ferrers board, i.e., the board obtained by removing the Ferrers diagram of a partition from the top right corner of an $n\times n$ chessboard. We consider a Markov chain on the set $R$ of rook placements on $B$ in which you can move from one placement to any other legal placement obtained by switching the columns in which two rooks sit. We give sharp estimates for the rate of convergence of this Markov chain using spectral methods. As part of this analysis we give a complete combinatorial description of the eigenvalues of the transition matrix for this chain. We show that two extremes cases of this Markov chain correspond to random walks on groups which are analyzed in the literature. Our estimates for rates of convergence interpolate between those two results.

scholarly journals Gillis' Random Walks on Graphs

2005 ◽  
Vol 42 (1) ◽  
pp. 295-301 ◽  
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Random Walks ◽  
Regular Graph ◽  
Simple Random Walk ◽  
Square Lattice ◽  
Random Walker ◽  
Unit Step ◽  
Random Walks On Graphs ◽  
The Green Function

We consider a random walker on a d-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of the d directions, with common probability 1/d for each one. At any later step, the random walker moves in any one of the directions, with probability q for a reversal of direction and probability p for any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is a d-dimensional square lattice. We prove that the Gillis random walk on a d-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.

scholarly journals Gillis' Random Walks on Graphs

2005 ◽  
Vol 42 (01) ◽  
pp. 295-301 ◽  
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Random Walks ◽  
Regular Graph ◽  
Simple Random Walk ◽  
Square Lattice ◽  
Random Walker ◽  
Unit Step ◽  
Random Walks On Graphs ◽  
The Green Function

We consider a random walker on ad-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of theddirections, with common probability 1/dfor each one. At any later step, the random walker moves in any one of the directions, with probabilityqfor a reversal of direction and probabilitypfor any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is ad-dimensional square lattice. We prove that the Gillis random walk on ad-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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