scholarly journals Computing Kemeny's constant for a barbell graph

2019 ◽  
Vol 35 ◽  
pp. 583-598 ◽  
Author(s):  
Jane Breen ◽  
Steve Butler ◽  
Nicklas Day ◽  
Colt DeArmond ◽  
Kate Lorenzen ◽  
...  
Keyword(s):  
Graph Theory ◽  
Random Walk ◽  
Weighted Average ◽  
Spectral Graph Theory ◽  
Explicit Computation ◽  
First Passage ◽  
Mean First Passage Times ◽  
Spectral Graph ◽  
The Mean ◽  
Passage Times

In a graph theory setting, Kemeny’s constant is a graph parameter which measures a weighted average of the mean first passage times in a random walk on the vertices of the graph. In one sense, Kemeny’s constant is a measure of how well the graph is ‘connected’. An explicit computation for this parameter is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length, as well as for two cliques joined by two paths of different length. In each case, Kemeny’s constant is shown to be O(n3), which is the largest possible order of Kemeny’s constant for a graph on n vertices. The approach used is based on interesting techniques in spectral graph theory and includes a generalization of using twin subgraphs to find the spectrum of a graph.

scholarly journals SIMPLE PROCEDURES FOR FINDING MEAN FIRST PASSAGE TIMES IN MARKOV CHAINS

2007 ◽  
Vol 24 (06) ◽  
pp. 813-829 ◽  
Author(s):  
JEFFREY J. HUNTER
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Linear Equations ◽  
First Passage Times ◽  
First Passage ◽  
Mean First Passage Times ◽  
The Mean ◽  
Computational Procedures ◽  
Passage Times

The derivation of mean first passage times in Markov chains involves the solution of a family of linear equations. By exploring the solution of a related set of equations, using suitable generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, we are able to derive elegant new results for finding the mean first passage times. As a by-product we derive the stationary distribution of the Markov chain without the necessity of any further computational procedures. Standard techniques in the literature, using for example Kemeny and Snell's fundamental matrix Z, require the initial derivation of the stationary distribution followed by the computation of Z, the inverse of I - P + eπT where eT = (1, 1, …, 1) and πT is the stationary probability vector. The procedures of this paper involve only the derivation of the inverse of a matrix of simple structure, based upon known characteristics of the Markov chain together with simple elementary vectors. No prior computations are required. Various possible families of matrices are explored leading to different related procedures.

Weighted trapping time of weighted directed treelike network

2020 ◽  
Vol 31 (08) ◽  
pp. 2050108
Author(s):  
Meifeng Dai ◽  
Yongbo Hou ◽  
Tingting Ju ◽  
Changxi Dai ◽  
Yu Sun ◽  
...  
Keyword(s):  
Complex Networks ◽  
First Passage Times ◽  
First Passage ◽  
Trade Network ◽  
Trapping Time ◽  
Undirected Network ◽  
Mean First Passage Times ◽  
Central Node ◽  
The Mean ◽  
Passage Times

With the deepening of research on complex networks, many properties of complex networks are gradually studied, for example, the mean first-passage times, the average receive times and the trapping times. In this paper, we further study the average trapping time of the weighted directed treelike network constructed by an iterative way. Firstly, we introduce our model inspired by trade network, each edge [Formula: see text] in undirected network is replaced by two directed edges with weights [Formula: see text] and [Formula: see text]. Then, the trap located at central node, we calculate the weighted directed trapping time (WDTT) and the average weighted directed trapping time (AWDTT). Remarkably, the WDTT has different formulas for even generations and odd generations. Finally, we analyze different cases for weight factors of weighted directed treelike network.

First passage time for the state of exact chemical equilibrium

1980 ◽  
Vol 45 (3) ◽  
pp. 777-782 ◽  
Author(s):  
Milan Šolc
Keyword(s):  
Chemical Equilibrium ◽  
First Passage Time ◽  
Passage Time ◽  
The State ◽  
Recurrence Time ◽  
First Passage ◽  
First Order ◽  
The Mean ◽  
Passage Times ◽  
Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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