Opinion Evolution in Divided Community

Abstract

Our agent-based model of opinion dynamics concerns the current vast divisions in modern societies. It examines the process of social polarization, understood here as the partition of a community into two opposing groups with contradictory opinions. Our goal is to measure how mutual animosities between parties may lead to their radicalization. We apply a double-clique topology with both positive and negative ties to the model of binary opinions. Individuals are subject to social pressure; they conform to the opinions of their own clique (positive links) and oppose those from the other one (negative links). There is also a chance of acting independently, which alters the system's behavior in various ways, depending on its magnitude. The results, obtained with both Monte-Carlo simulations and the mean-field approach, lead to two main conclusions: in such a system, there exists a critical quantity of negative relations that are needed for polarization to occur, and (rather surprisingly) independent actions actually support the process, unless their frequency is too high, in which case the system falls into total disorder.

Keywords: Monte-Carlo simulation; agent-based model; opinion dynamics; social polarization.

Figure 1

Schematic representation of the opinion update of a single agent that was initially in the up state. With probability h, the agent acts independently and changes opinion randomly. With complementary probability $1-p$, the agent is subject to social influence.

Figure 2

All possible choices of the influence group in the model with $q=4$ that lead to an opinion flip by a target from clique A that was initially in state $S=-1$. The influence group may contain members from both cliques. Due to the presence of both positive and negative ties, the concept of unanimity from the original q-voter model has to be extended to signals, which are then received by the target of influence. A signal is the state of a member when coming from target’s clique, or its inverted state otherwise. The target changes its opinion only if all members of the influence group emit the same signal.

Figure 3

The annealed model: direction fields of the model described by Equation (11) with fixed points marked with circles for different values of independence h and two values of parameter p, $0.1$ (left column) and $0.2$ (right column).

Figure 4

Comparison between the quenched (left column) and annealed (right column) models: product of magnetizations $m_A{m}_B$ as a function of time and L, for two different independence levels, $h=0,0.1$.

Figure 5

Comparison between the quenched (left column) and annealed (right column) models: product of magnetizations $m_A{m}_B$ as a function of time and h, for two different fractions of cross-links, $L=0.3,0.6$.

Figure 6

Comparison between the quenched (left) and annealed (right) models: final product of magnetizations $m_A{m}_B$ (top) and its projection on the $(h,L)$ plane (bottom). The blue, red and purple colors correspond to consensus, polarization and disorder, respectively. In both models, we can observe that the critical value $L^{*}$ decreases with an increase in h, while L has only a marginal impact on $h^{*}$.