# Homogeneous symmetrical threshold model with nonconformity: independence vs. anticonformity

## TL;DR

This paper compares two variants of a modified Watts threshold model with nonconformity on a complete graph, analyzing how independence and anticonformity influence phase transitions and order-disorder behavior.

## Contribution

It introduces and analyzes two nonconformity variants of the homogeneous Watts threshold model, revealing their similar behavior at r=0.5 and differences for r>0.5.

## Key findings

- At r=0.5, the transition is continuous.
- Models with independence and anticonformity produce similar results, scaled by a factor of 2.
- For r>0.5, the independence model exhibits a discontinuous transition and hysteresis.

## Abstract

We study two variants of the modified Watts threshold model with a noise (with nonconformity, in the terminology of social psychology) on a complete graph. Within the first version, a noise is introduced via so-called independence, whereas in the second version anticonformity plays the role of a noise, which destroys the order. The modified Watts threshold model, studied here, is homogeneous and posses an up-down symmetry, which makes it similar to other binary opinion models with a single-flip dynamics, such as the majority-vote and the q-voter models. Because within the majority-vote model with independence only continuous phase transitions are observed, whereas within the q-voter model with independence also discontinuous phase transitions are possible, we ask the question about the factor, which could be responsible for discontinuity of the order parameter. We investigate the model via the mean-field approach, which gives the exact result in the case of a complete graph, as well as via Monte Carlo simulations. Additionally, we provide a heuristic reasoning, which explains observed phenomena. We show that indeed, if the threshold r = 0.5, which corresponds to the majority-vote model, an order-disorder transition is continuous. Moreover, results obtained for both versions of the model (one with independence and the second one with anticonformity) give the same results, only rescaled by the factor of 2. However, for r > 0.5 the jump of the order parameter and the hysteresis is observed for the model with independence, and both versions of the model give qualitatively different results.