Abstract
This chapter is devoted to the analysis of a simple opinion model in order to illustrate the ideas developed in the previous chapter. The projection from micro to macro emphasizes the particular role played by homogeneous mixing as a requirement for the Markovianity of the projected model. We present a Markov chain analysis of the binary voter model (VM) with a particular focus on its transient dynamics and show that the general VM can be reduced to the binary case by further projection. Finally, the question of interaction constraints in form of bounded confidence is addressed. Homogeneous interaction probabilities (homogeneous mixing) are assumed in all the analyses presented in this chapter. Interaction heterogeneities are left for the next chapters.
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Notes
- 1.
Notice that permutation invariance is also present if “self interactions” are excluded such that ω(i, i) = 0. Then \(\omega (i,j) = 1/N(N - 1),\forall i\neq j\). For the following computations the possibility that an agent i “interacts” with itself is not excluded.
- 2.
Alternatively, one could also verify the lumpability of Y directly with respect to the micro process. Namely, as shown in Sect. 3.2, the micro chain is a random walk on H(N, δ). The group \(\mathcal{S}_{N}\) acting on the agents as well as the permutation group \(\mathcal{S}_{\delta }\) acting on the agent attributes give rise to automorphisms of H(N, δ) such that the automorphism group is given by the direct product \(Aut(H(N,\delta )) = \mathcal{S}_{N} \otimes \mathcal{S}_{\delta }\). The transformation group that generates the new partition Y is a subgroup of that, namely, \(\varLambda = \mathcal{S}_{N} \otimes \mathcal{S}_{\delta -1} \subset \mathcal{S}_{N} \otimes \mathcal{S}_{\delta }\).
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Banisch, S. (2016). The Voter Model with Homogeneous Mixing. In: Markov Chain Aggregation for Agent-Based Models. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-24877-6_4
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