Abstract
Spin dynamics on networks allows us to understand how a global consensus emerges out of individual opinions. Here, we are interested in the effect of heterogeneity in the initial geographic distribution of a competing opinion on the competitiveness of its own opinion. Accordingly, in this work, we studied the effect of spatial heterogeneity on the majority rule dynamics using a three-state spin model, in which one state is neutral. Monte Carlo simulations were performed on square lattices divided into square blocks (cells). Accordingly, one competing opinion was distributed uniformly among cells, whereas the spatial distribution of the rival opinion was varied from the uniform to heterogeneous, with the median-to-mean ratio in the range from 1 to 0. When the size of discussion group is odd, the uncommitted agents disappear completely after 3.30 ± 0.05 update cycles, and then the system evolves in a two-state regime with complementary spatial distributions of two competing opinions. Even so, the initial heterogeneity in the spatial distribution of one of the competing opinions causes a decrease of this opinion competitiveness. That is, the opinion with initially heterogeneous spatial distribution has less probability to win, than the opinion with the initially uniform spatial distribution, even when the initial concentrations of both opinions are equal. We found that although the time to consensus , the opinion’s recession rate is determined during the first 3.3 update cycles. On the other hand, we found that the initial heterogeneity of the opinion spatial distribution assists the formation of quasi-stable regions, in which this opinion is dominant. The results of Monte Carlo simulations are discussed with regard to the electoral competition of political parties.
Highlights
Nowadays, it is recognized that many features of collective behavior of large social systems are almost independent of the attributes of individuals and details of social interactions [1]
We found that the time to transition from the three to two -state regime increases linearly, with respect to the number of nodes N and it is independent of the initial spatial distributions of non-zero spins
This means that our majority rule model evolves in the three-state regime during only the first Ttr = ttr / N = 3.30 ± 0.05 cycles of N updates, independently of the lattice size and heterogeneity in the initial spatial distributions of the rival opinions
Summary
It is recognized that many features of collective behavior of large social systems are almost independent of the attributes of individuals and details of social interactions [1]. This gives rise to the use of spin models for modeling opinion dynamics in complex social systems [1,2,3,4,5]. A spin model consists of N spins located on the network nodes. Each node can adopt any of two or more admissible spin states defined in the model. In the context of opinion dynamics, spin states can be associated with competing opinions and uncommitted attitudes
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