Testing the validity of the Kirkwood approximation using an extended Sznajd model

TL;DR

This study evaluates the Kirkwood approximation's accuracy in predicting the exit probability of an extended Sznajd model, revealing its strengths and limitations across different parameter regimes through simulations and analytical comparisons.

Contribution

It extends the Sznajd model to include multiple opinion dynamics and tests the Kirkwood approximation's validity in this broader context.

Findings

Kirkwood approximation accurately predicts exit probabilities in certain parameter regions.

The approximation fails in some regimes, highlighting its limitations.

Simulation results help delineate the approximation's reliability boundaries.

Abstract

We revisit the deduction of the exit probability of the one dimensional Sznajd model through the Kirkwood approximation [F. Slanina et al., Europhys. Lett. 82, 18006 (2008)]. This approximation is peculiar in that in spite of the agreement with simulation results [F. Slanina et al., Europhys. Lett. 82, 18006 (2008), R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008), A. M. Timpanaro and C. P. C. Prado, 89, 052808 (2014)] the hypothesis about the correlation lenghts behind it are inconsistent and fixing these inconsistencies leads to the same results as a simple mean field. We use an extended version of the Sznajd model to test the Kirkwood approximation in a wider context. This model includes the voter, Sznajd and "United we Stand, Divided we Fall" (USDF) models [R. A. Holley and T. M. Liggett, Ann. Prob. 3, 643 (1975), K. Sznajd-Weron and J. Sznajd, Int. Journ. Mod. Phys. C 11, 1157 (2000)] as different parameter combinations, meaning that some analytical results from these models can be used to evaluate the performance of the Kirkwood approximation. We also compare the predicted exit probability with simulation results for networks with $10^3$ sites. The results show clearly the regions in parameter space where the approximation gives accurate predictions, as well as where it starts failing, leading to a better understanding of its reliability.