The threshold model with anticonformity under random sequential updating

TL;DR

This paper analyzes an asymmetric threshold model with anticonformity on complete graphs, using multiple approaches to understand social dynamics like hysteresis and critical mass, especially in heterogeneous populations.

Contribution

It introduces an analytical framework combining mean-field, Monte Carlo, and Markov chain methods for the model, highlighting behaviors in heterogeneous social systems.

Findings

Large systems show consistent results across approaches.

Heterogeneous thresholds lead to social hysteresis and critical mass phenomena.

Distribution shape influences complex social behaviors.

Abstract

We study an asymmetric version of the threshold model with anticonformity under asynchronous update mode that mimics continuous time. We study this model on a complete graph using three different approaches: mean-field approximation, Monte Carlo simulation, and the Markov chain approach. The latter approach yields analytical results for arbitrarily small systems, in contrast to the mean-field approach, which is strictly correct only for an infinite system. We show that for sufficiently large systems, all three approaches produce the same results, as expected. We consider two cases: (1) homogeneous, in which all agents have the same tolerance threshold, and (2) heterogeneous, in which the thresholds are given by a beta distribution parametrized by two positive shape parameters $\alpha$ and $\beta$. The heterogeneous case can be treated as a generalized model that reduces to a homogeneous model in special cases. We show that particularly interesting behaviors, including social hysteresis and critical mass, arise only for values of $\alpha$ and $\beta$ that yield the shape of the distribution observed in real social systems.