GRAPH INTERPRETATION OF IRREDUCIBLE, REDUCIBLE, PERIODIC, AND APERIODIC PROPERTIES IN MARKOV CHAINS - Suci Rachmadini, Haliza; Muhammad, Faisal; Razvan Serban (Journal of Mathematics and Scientific Computing With Applications, 2025)
Markov chains are widely used stochastic models for describing dynamic systems whose future states depend only on short-term probabilistic transitions. Key structural properties irreducibility, reducibility, periodicity, and aperiodicity are crucial for understanding long-term behavior, particularly the existence and stability of stationary distributions. Traditionally, these characteristics are determined through analysis of the transition probability matrix; however, this approach can be computationally demanding and difficult to interpret for large systems. This study explores an alternative representation using directed graphs, where each state is modeled as a node and each positive transition probability as a directed edge. The approach connects irreducibility with strong graph connectivity, while reducibility corresponds to the presence of separate communication classes. Periodicity and aperiodicity are identified through the structure of cycles and the greatest common divisor of return path lengths. The results demonstrate that graph-based analysis provides clearer and more intuitive framework for examining structural properties of Markov chains.