Abstract
In the third chapter, we have seen that an agent-based model (ABM) defines a process of change at the individual level—a micro process—by which in each time step one configuration of individuals is transformed into another configuration. For a class of models we have shown this micro process to be a Markov chain on the space of all possible agent configurations. Moreover, we have shown that the full aggregation—that is, the re-formulation of the model by mere aggregation over the individual attributes of all agents—may give rise to a new process that is again a Markov chain, however, only under the rather restrictive assumption of homogeneous mixing. Heterogeneities in the micro description, in general, destroy the Markov property of the macro process obtained by such a full aggregation.
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Notes
- 1.
I am grateful to an anonymous reviewer for this formulation.
- 2.
Notice that the case M = L is special because it leads to additional symmetries as the two communities are interchangeable. This is not generally the case and therefore we develop the more general case of M ≠ L here, even if the computations are mostly performed for the example \(M = L = 50\).
- 3.
The reason for this is clear. The number of micro configurations x ∈ Σ mapped into the state \(\tilde{X}_{m,l}\) is \(\binom{M}{m}\binom{L}{l}\) which is a huge number for \(m \approx M/2,l \approx L/2\) but only 1 for \(m = M,l = 0\) and \(m = 0,l = L\). Because under homogeneous mixing there is no favoring of particular agent configurations with the same \(k = m + l\) the stationary probability at macro scale is proportional to the cardinality of the set \(\tilde{X}_{m,l}\).
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Banisch, S. (2016). From Network Symmetries to Markov Projections. In: Markov Chain Aggregation for Agent-Based Models. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-24877-6_5
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