scholarly journals On the evolution of topology in dynamic clique complexes

2016 ◽  
Vol 48 (4) ◽  
pp. 989-1014 ◽  
Author(s):  
Gugan C. Thoppe ◽  
D. Yogeshwaran ◽  
Robert J. Adler
Keyword(s):  
Markov Chains ◽  
Betti Number ◽  
Continuous Time ◽  
Time Varying ◽  
Inclusion Probability ◽  
Uhlenbeck Process ◽  
Continuous Time Markov Chains ◽  
Ornstein Uhlenbeck Process

AbstractWe consider a time varying analogue of the Erdős–Rényi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous-time Markov chains. Our main result is that when the edge inclusion probability is of the form p=nα, where n is the number of vertices and α∈(-1/k, -1/(k + 1)), then the process of the normalised kth Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process as n→∞.

scholarly journals Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 42
Author(s):  
Alexander Zeifman ◽  
Yacov Satin ◽  
Ivan Kovalev ◽  
Rostislav Razumchik ◽  
Victor Korolev
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Numerical Solutions ◽  
Time Interval ◽  
Single Server ◽  
Time Varying ◽  
Logarithmic Norm ◽  
Discrete State ◽  
Continuous Time Markov Chains

The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose the position and the length of the “distant time interval” (in the periodic case) on which the solution has to be computed. They can also be helpful whenever the state space truncation is required. In this paper one such analytic method—the logarithmic norm method—is being reviewed. Its applicability is shown within the queueing theory context with three examples: the classical time-varying M/M/2 queue; the time-varying single-server Markovian system with bulk arrivals, queue skipping policy and catastrophes; and the time-varying Markovian bulk-arrival and bulk-service system with state-dependent control. In each case it is shown whether and how the bounds on the rate of convergence can be obtained. Numerical examples are provided.

scholarly journals Trace Machines for Observing Continuous-Time Markov Chains

2006 ◽  
Vol 153 (2) ◽  
pp. 259-277 ◽  
Author(s):  
Verena Wolf ◽  
Christel Baier ◽  
Mila Majster-Cederbaum
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Continuous Time Markov Chains

Bounds and Approximations for the Transient Behavior of Continuous-Time Markov Chains

1989 ◽  
Vol 3 (2) ◽  
pp. 175-198 ◽  
Author(s):  
Bok Sik Yoon ◽  
J. George Shanthikumar
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
First Passage Time ◽  
Transient Behavior ◽  
Passage Time ◽  
Lower And Upper Bounds ◽  
Simple Method ◽  
Continuous Time Markov Chains ◽  
Recent Addition ◽  
Dependent Probability

Discretization is a simple, yet powerful tool in obtaining time-dependent probability distribution of continuous-time Markov chains. One of the most commonly used approaches is uniformization. A recent addition to such approaches is an external uniformization technique. In this paper, we briefly review these different approaches, propose some new approaches, and discuss their performances based on theoretical bounds and empirical computational results. A simple method to get lower and upper bounds for first passage time distribution is also proposed.

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