TRANSFORMATION OF COMPARTMENT MODELS INTO 1D RANDOM WALKS ON WEIGHTED DIGRAPHS
DOI:
https://doi.org/10.25077/jmua.15.4.544-550.2026Keywords:
1D Random Walk, Discrete-Time Markov Chain, Metzler Matrix, Network Combinatorics, Weighted DigraphAbstract
This study constructs a transformation of a continuous dynamic compartment model into a Discrete-Time Markov Chain (DTMC) based on a onedimensional (1D) random walk, specifically emphasizing the combinatoric properties of a weighted digraph. The network’s combinatoric characteristics are analyzed through vertex decomposition of the compartments and the mapping of a weighted adjacency matrix to ensure flow conservation across each directed edge. The stability analysis of the system’s unique equilibrium point is conducted using network topology approaches via the Cyclic Small-Gain Theorem for Metzler matrices and spectral localization using Gershgorin’s Disc Theorem. The results demonstrate a robust structural consistency, where the combinatoric loop gain of the graph strictly ranges between 0 and 1, acting as a natural damping mechanism for the population. Furthermore, the stationary distribution of the micro-level random walk precisely converges to the macro-level equilibrium ratio, confirming a strict mathematical link between the combinatoric structure of the weighted graph and the stochastic behavior of the system.
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