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

Abstract

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.

Pair approximation for the q-voter models with quenched disorder on networks

“…Especially prone to this kind of error is the q-voter model with anticonformity. Similar discrepancies are reported with regard to other dynamics [28,29,32,33,35,36]. However, we have not observed such errors for the quenched counterparts of the studied models.…”

“…The pair approximation is a general technique used to study various dynamics on static [28][29][30][31][32][33][34][35][36][37][38][39][40] as well as coevolutionary networks [41][42][43][44][45]. This method has already been applied to the q-voter models with nonconformity under the annealed approach.…”

“…[30-32, 34, 35], whereas for the q-voter model with anticonformity in Refs. [33,36]. In this section, we develop the pair approximation for the quenched versions of the q-voter models.…”

“…The question which naturally arises is whether the above results remain valid on networks. Although both the models have been studied extensively under the annealed approach on networks, by the use of the pair approximation and Monte Carlo simulations [30][31][32][33][34][35][36], such a comprehensive analysis is missing for their quenched counterparts.…”

A general framework to link theory and empirics in opinion formation models

“…Besides, the noisy Voter model assumes that agents, regardless of their surroundings, may spontaneously change their views 34 , 39 . More complex opinion dynamics protocols can be found in the Majority rule 41 , Sznajd 17 , or q-Voter 38 , 42 models.…”