Artifacts of opinion dynamics at one dimension

TL;DR

This paper analyzes opinion dynamics in a one-dimensional Ising model using three local update rules, showing that different formulas for final state probabilities are identical and discussing implications for social behavior modeling.

Contribution

It provides exact calculations of final state probabilities for three local update rules and compares them to Kirkwood approximation results, clarifying discrepancies and artifacts.

Findings

Exact formulas match Kirkwood approximation results.

Discrepancies with Galam unifying frame are due to implementation differences.

Non stepwise exit probability is an artifact of finite size systems.

Abstract

The dynamics of a one dimensional Ising spin system is investigated using three families of local update rules, the Galam majority rules, Glauber inflow influences and Sznadj outflow drives. Given an initial density p of up spins the probability to reach a final state with all spins up is calculated exactly for each choice. The various formulas are compared to a series of previous calculations obtained analytically using the Kirkwood approximation. They turn out to be identical. The apparent discrepancy with the Galam unifying frame is addressed. The difference in the results seems to stem directly from the implementation of the local update rule used to perform the associated numerical simulations. The findings lead to view the non stepwise exit probability as an artifact of the one dimensional finite size system with fixed spins. The suitability and the significance to perform numerical simulations to model social behavior without solid constraints is discussed and the question of what it means to have a mean field result in this context is addressed.