Emergence of opinion splits in the Sznajd model with latency

TL;DR

This work examines what happens when latency is added to the Sznajd model, and finds that for low latency, the model behaves roughly like it does in the absence of latency, where one opinion will always eventually dominate.

Abstract

In the modelling of social systems, opinion latency is the idea that once an agent changes its opinion, there will be a period of time where it is immune to other changes. When added to the voter model this leads to a situation where no matter how low the latency is or how many opinions are considered, all opinions end up in a coexistence where they are equally represented. In this work, we examine what happens when latency is added to the Sznajd model. What we find is that for low latency, the model behaves roughly like it does in the absence of latency, where one opinion will always eventually dominate. For high latency, the possibility for a symmetric coexistence of 2 opinions arises, but contrary to the voter model, a coexistence of more than 2 opinions is never stable. We provide evidence of this phenomenon with computer simulations of the model in Barabási-Albert networks, together with a mean field treatment that is able to capture the observed behavior. We argue that this could hint at an explanation for the prevalence of two opinion splits in the real world.

Figures (5)

• Figure 1: Average over $10^3$ simulations of the number of opinions present in the network as a function of time (measured in MCTs). The simulations were done in both versions of the Sznajd model with latency (inflow and outflow), for starting amounts of opinion $M_0 = 3,4,5,8, 12$ and $p$ ranging from $0.03$ to $0.3$ ($0.06, 0.12$ and $0.18$ shown in the figure).
• Figure 2: Timeseries of the amount of opinions present in the network for $10^3$ different simulations, starting with 3 different opinions and using $p=0.03$. For the time unit, 1 kMCT = $10^3$ MCTs. (a) Inflow Sznajd model. (b) Outflow Sznajd model.
• Figure 3: Proportion of simulations that ended up in a consensus state, starting with 2 different opinions, as a function of the parameter $p$ and of the asymmetry $\Delta = |\theta_{1\circ} - \theta_{2\circ}|$, where $\theta_{1(2)\circ}$ is the starting proportion of agents with opinion 1 (2). For each point in the graph, 100 simulations were made with a duration of $10^3$ MCTs. (a) Inflow Sznajd model. (b) Outflow Sznajd model.
• Figure 4: Evolution of $\theta_{\mathrm{max}}$, the proportion of agents holding the most common opinion as a function of time. Simulations were made for the voter model with latency using $p=1$ and different initial amounts of opinions $M=2,3,5,8$. Denoting the starting proportion of agents holding opinion $\sigma$ by $\theta_{\sigma}$, the exact initial conditions were: $\theta_1 = 0.8$, $\theta_2 = 0.2$ for $M=2$; $\theta_1 = 0.8$, $\theta_2 = \theta_3 = 0.1$ for $M=3$; $\theta_1 = 0.8$, $\theta_2 = \ldots = \theta_5 = 0.05$ for $M=5$ and $\theta_1 = 0.86$, $\theta_2 = \ldots = \theta_8 = 0.02$ for $M=8$. Each band covers from one standard deviation below the average $\theta_{\max}$ to one standard deviation above the average. For each value of $M$, $10^3$ simulations were made. After 500 MCTs, all of the simulations still had all $M$ opinions present.
• Figure 5: Parameters for which the mean field equations of the Sznajd model converge to consensus (yellow) or to a symmetric coexistence (purple) as a function of $\lambda$ and of the asymmetry $\Delta = |\theta_{1\circ} - \theta_{2\circ}|$, where $\theta_{1(2)\circ}$ is the starting proportion of agents with opinion 1 (2). (a) With the initial condition such that all agents are active, matching what was done in the simulations. (b) With the initial condition such that all agents are latent.