Abstract
This chapter is devoted to the study of a non-Markovian case building upon the analysis of the contrarian voter model (CVM) discussed in the previous chapter. As noted earlier, two things may happen by projecting the microscopic Markov chain associated to an agent-based model (ABM) onto a coarser partition. First, the macro process is still a Markov chain which is the case of lumpability discussed most extensively throughout this book. Secondly, Markovianity may be lost after the projection induced by a certain observable which means that memory effects are introduced at the macroscopic level. This is a fingerprint of emergence in models of self-organizing systems. Noteworthy, in ABMs as well as more generally in Markov chains, this situation is the rule rather than an exception (Chazottes and Ugalde 2003; Gurvits and Ledoux 2005; Banisch et al. 2012; Banisch 2014).
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Notes
- 1.
This example will be explored carefully in future work. For now, just notice that the VM on the ring leads to a long-lasting pattern of a single white and a single black regime and further change in the number of white and black agents only happens when an edge at the interface between the two regimes is chosen. The respective probability is equal for all micro configurations of this kind and other (disordered) configurations are not visited once such a situation has been reached. See also Fig. 1.1 in the Introduction.
- 2.
Notice that the number of possibilities reduces at the corners or borders of the meso chain whenever m = 0 or l = 0.
- 3.
This is to avoid a possible ambiguity because the probability \(Pr[X_{k}\vert \tilde{X}_{m,l}]\) could also be read in terms of the projection from \(\tilde{\mathbf{X}}\) to X where \(Pr[X_{k}\vert \tilde{X}_{m,l}]\) would indicate the probability with which the meso state \(\tilde{X}_{m,l}\) is taken by ϕ to the macro state X k .
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Banisch, S. (2016). Information-Theoretic Measures for the Non-Markovian Case. In: Markov Chain Aggregation for Agent-Based Models. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-24877-6_7
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