A Study on Markov Chain Properties in Terms of Cliques

Abstract

A probabilistic stochastic model deals with the real life applications. An important class of stochastic processes is the Markov process which possesses the past forgetting property. As the Markov process (also called Markov chain) is a memoryless process, it is more advantageous to work with the systems with less number of parameters. The situation and the things happening around us are always probabilistic and non-deterministic. The model which works efficiently on probabilistic conditions is the Markov Chain. Each chain has its own identity and applications in and around the real world problems. newlineThe Markov chain can be categorized as finite Markov chain, infinite Markov chain based on the number of States. If the Markov chain is finite thenit has finite number of states, the probabilities exchanged between the states are fixed and the states attain their steady state in the course of time. newlineSince the transition can start at any state of the Markov chain, knowing the properties of the Markov chain gives insight for various analyses. This Markov property enables reasoning and computation with the model that would be otherwise intractable. Unless the Properties of the states are not analyzed the long run behavior and stability of the Markov chain cannot be determined. newlineWhen the states of the Markov chain are large in number, discussions on the properties of the Markov chain using probabilistic method is tedious and laborious. Traditional probabilistic methods for examining the properties of Markov chains are often complex and time-consuming. By transforming the problem into a graph-theoretical newlineframework, we aim to simplify the analysis and provide new insights into the structural and dynamic characteristics of Markov chains. newlineIn this research work, the properties of the Markov chain are discussed and analyzed using different types of graphs according to the nature of the Markov Chain. Hence we tried to form a bridge for the properties of the Markov chain using graph theory in terms of cliques.