We study the binary q-voter model with generalized anticonformity on random Erdős–Rényi 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 $$q_c$$ q c in case of conformity is independent from the size of the source of influence $$q_a$$ q a in case of anticonformity. For $$q_c=q_a=q$$ 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 $$q_c \ge q_a + \Delta q$$ q c ≥ q a + Δ q , where $$\Delta q=4$$ Δ q = 4 for $$q_a \le 3$$ q a ≤ 3 and $$\Delta q=3$$ Δ q = 3 for $$q_a>3$$ q a > 3 . In this paper, we pose the question if discontinuous phase transitions survive on random graphs with an average node degree $$\langle k\rangle \le 150$$ ⟨ k ⟩ ≤ 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 $$\langle k\rangle$$ ⟨ k ⟩ . Moreover, we show that for $$q_a < q_c - 1$$ q a < q c - 1 pair approximation results overlap the Monte Carlo ones. On the other hand, for $$q_a \ge q_c - 1$$ q a ≥ 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 $$\langle k\rangle$$ ⟨ k ⟩ , as long as the pair approximation indicates correctly the type of the phase transition.

In this work, we address the question of the role of the influence of group size on the emergence of various collective social phenomena, such as consensus, polarization and social hysteresis. To answer this question, we study the three-state noisy q-voter model with bounded confidence, in which agents can be in one of three states: two extremes (leftist and rightist) and centrist. We study the model on a complete graph within the mean-field approach and show that, depending on the size q of the influence group, saddle-node bifurcation cascades of different length appear and different collective phenomena are possible. In particular, for all values of q>1, social hysteresis is observed. Furthermore, for small values of q∈(1,4), disagreement, polarization and domination of centrists (a consensus understood as the general agreement, not unanimity) can be achieved but not the domination of extremists. The latter is possible only for larger groups of influence. Finally, by comparing our model to others, we discuss how a small change in the rules at the microscopic level can dramatically change the macroscopic behavior of the model.

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