Markov Chain Aggregation for Agent-Based Models

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This self-contained text develops a Markov chain approach for analyzing microscopic models that describe complex systems' dynamics at the individual level. It introduces a general framework for aggregation in agent-based and related computational models, utilizing lumpability and information theory to connect micro and macro levels of observation. The foundation is a microscopic Markov chain representation that aligns with the dynamics of the agent-based model (ABM), treating all possible agent configurations as the state space of a large Markov chain. An explicit formal representation of a resulting "micro-chain," including microscopic transition rates, is derived using the random mapping representation of a Markov process. The choice of probability distribution for the stochastic component of the model is crucial, influencing the updating rule and dynamics at a Markovian level, particularly in "voter-like" models relevant to population genetics, evolutionary game theory, and social dynamics. The text illustrates that the aggregation problem in ABMs, especially lumpability conditions, can be integrated into a broader framework that employs information theory to identify various levels and relevant scales within complex dynamical systems.