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

TL;DR

This study investigates whether discontinuous phase transitions in a generalized q-voter model with anticonformity persist on random graphs similar to social networks, using simulations and pair approximation methods.

Contribution

It demonstrates that discontinuous phase transitions can occur on random graphs with moderate connectivity, extending previous mean-field results and analyzing the accuracy of approximation methods.

Findings

Discontinuous phase transitions survive on random graphs with average degree up to 150.

Pair approximation aligns with Monte Carlo results for certain parameter ranges.

The difference between spinodals follows a power law relative to average degree.

Abstract

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.