## Markov Chain Aggregation for Agent-Based ModelsThis self-contained text develops a Markov chain approach that makes the rigorous analysis of a class of microscopic models that specify the dynamics of complex systems at the individual level possible. It presents a general framework of aggregation in agent-based and related computational models, one which makes use of lumpability and information theory in order to link the micro and macro levels of observation. The starting point is a microscopic Markov chain description of the dynamical process in complete correspondence with the dynamical behavior of the agent-based model (ABM), which is obtained by considering the set of all possible agent configurations as the state space of a huge Markov chain. An explicit formal representation of a resulting “micro-chain” including microscopic transition rates is derived for a class of models by using the random mapping representation of a Markov process. The type of probability distribution used to implement the stochastic part of the model, which defines the updating rule and governs the dynamics at a Markovian level, plays a crucial part in the analysis of “voter-like” models used in population genetics, evolutionary game theory and social dynamics. The book demonstrates that the problem of aggregation in ABMs - and the lumpability conditions in particular - can be embedded into a more general framework that employs information theory in order to identify different levels and relevant scales in complex dynamical systems |

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### Contents

1 | |

11 | |

3 AgentBased Models as Markov Chains | 35 |

4 The Voter Model with Homogeneous Mixing | 56 |

5 From Network Symmetries to Markov Projections | 83 |

6 Application to the Contrarian Voter Model | 109 |

7 InformationTheoretic Measures for the NonMarkovian Case | 127 |

8 Overlapping Versus Nonoverlapping Generations | 156 |

A Synthesis | 177 |

10 Conclusion | 187 |

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### Common terms and phrases

ABMs absorbing agent configurations Agent-Based Models analysis assortative mating attributes Banisch behavior binary bounded confidence chapter complete graph Complex Systems computational consensus consider contrarian rate convergence corresponding defined denote dynamical systems emergence evolution finite fitness landscape full aggregation fundamental matrix global Görnerup and Jacobi Hamming weight heterogeneity homogeneous mixing individuals initial interaction topology Kemeny and Snell leader leads lumpability lumpable with respect macro level macro process macroscopic Markov chain Markov Chain Aggregation Markovianity measure mathematical micro chain micro configurations mutual information networks nodes Notice number of agents observed opinion dynamics partition Pfante polarization population possible probability distribution projection QXm;l random walk regular graphs relation replacement rules Sect Shalizi simulation social speciation Springer stationary distribution stochastic structure symmetries system property Theorem transient transition matrix transition probabilities update vector voter model weak emergence weak lumpability WF model Wimsatt ω ω