Phase transitions in the q-voter model with two types of stochastic driving

In this paper we study nonlinear $q$-voter model with stochastic driving on a complete graph. We investigate two types of stochasticity that, using the language of social sciences, can be interpreted as different kinds of nonconformity. From a social point of view, it is very important to distinguish between two types nonconformity, so called anti-conformity and independence. A majority of works suggests that these social differences may be completely irrelevant in terms of microscopic modeling that uses tools of statistical physics and that both types of nonconformity play the role of so called 'social temperature'. In this paper we clarify the concept of 'social temperature' and show that different type of 'noise' may lead to qualitatively different emergent properties. In particularly, we show that in the model with anti-conformity the critical value of noise increases with parameter $q$, whereas in the model with independence the critical value of noise decreases with the $q$. Moreover, in the model with anti-conformity the phase transition is continuous for any value of $q$, whereas in the model with independence the transition is continuous for $q \le 5$ and discontinuous for $q>5$.