By Sven Banisch
This self-contained textual content develops a Markov chain strategy that makes the rigorous research of a category of microscopic versions that designate the dynamics of complicated platforms on the person point attainable. It offers a common framework of aggregation in agent-based and similar computational versions, one that uses lumpability and data concept on the way to hyperlink the micro and macro degrees of remark. the start line is a microscopic Markov chain description of the dynamical technique in entire correspondence with the dynamical habit of the agent-based version (ABM), that's received by way of contemplating the set of all attainable agent configurations because the kingdom house of an important Markov chain. An specific formal illustration of a ensuing “micro-chain” together with microscopic transition premiums is derived for a category of versions through the use of the random mapping illustration of a Markov method. the kind of likelihood distribution used to enforce the stochastic a part of the version, which defines the updating rule and governs the dynamics at a Markovian point, performs a very important half within the research of “voter-like” types utilized in inhabitants genetics, evolutionary online game concept and social dynamics. The booklet demonstrates that the matter of aggregation in ABMs - and the lumpability stipulations specifically - should be embedded right into a extra common framework that employs details idea with a purpose to determine diverse degrees and appropriate scales in advanced dynamical systems
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Additional resources for Markov Chain Aggregation for Agent-Based Models
Example text
Levin, S. , & Andreasen, V. (1989). Aggregation in model ecosystems II. Approximate aggregation. Mathematical Medicine and Biology, 6(1), 1–23. Izquierdo, L. , Izquierdo, S. , Galán, J. , & Santos, J. I. (2009). Techniques to understand computer simulations: Markov chain analysis. Journal of Artificial Societies and Social Simulation, 12(1), 6. Jacobi, M. N. (2008). A dual eigenvector condition for strong lumpability of Markov chains. In CoRR. 1986v2. Jacobi, M. , & Görnerup, O. (2009). A spectral method for aggregating variables in linear dynamical systems with application to cellular automata renormalization.
2011, and references therein). The main idea 24 2 Background and Concepts in computational mechanics is to group histories which give rise to the same conditional probability distribution over futures into equivalence classes—so-called causal states—and to construct in this way a minimal causal model—called -machines—for the prediction of the process at question. The reader may be referred to Shalizi and Crutchfield (2001) for an overview and several interesting theoretical results in computational mechanics.
The lumpability theory adopted for the purposes of this thesis is largely based on Kemeny and Snell (1976), which is, to the authors knowledge, the first textbook in which the strong as well as the weak form of lumpability are discussed with some detail. Notice that there are some other early and seminal works on lumpability, such as Burke and Rosenblatt (1958), Rosenblatt (1959), and Rogers and Pitman (1981). To illustrate the concept of strong lumpability, let us use the Land of Oz example repeatedly considered in Kemeny and Snell (1976) (see pages 29/30 for the introduction of the example and page 125 for the lumpability example).