Discontinuous phase transitions in the q-voter model with generalized anticonformity on random graphs

We study the binary $q$-voter model with generalized anticonformity on random Erd\H{o}s-R\'enyi graphs. In such a model, two types of social responses, conformity and anticonformity, occur with complementary probabilities and the size of the source of influence $q_c$ in case of conformity is independent from the size of the source of influence $q_a$ in case of anticonformity. For $q_c=q_a=q$ the model reduces to the original $q$-voter model with anticonformity. Previously, such a generalized model was studied only on the complete graph, which corresponds to the mean-field approach. It was shown that it can display discontinuous phase transitions for $q_c \ge q_a + \Delta q$, where $\Delta q=4$ for $q_a \le 3$ and $\Delta q=3$ for $q_a>3$. In this paper, we pose the question if discontinuous phase transitions survive on random graphs with an average node degree $\langle k\rangle \le 150$ observed empirically in social networks. Using the pair approximation, as well as Monte Carlo simulations, we show that discontinuous phase transitions indeed can survive, even for relatively small values of $\langle k\rangle$. Moreover, we show that for $q_a < q_c - 1$ pair approximation results overlap the Monte Carlo ones. On the other hand, for $q_a \ge q_c - 1$ pair approximation gives qualitatively wrong results indicating discontinuous phase transitions neither observed in the simulations nor within the mean-field approach. Finally, we report an intriguing result showing that the difference between the spinodals obtained within the pair approximation and the mean-field approach follows a power law with respect to $\langle k\rangle$, as long as the pair approximation indicates correctly the type of the phase transition.