Spontaneous Opinion Swings in the Voter Model with Latency

TL;DR

The problem addressed is how to capture reluctance to change opinions in a binary-choice voter model. The authors introduce a constant latency after a flip, producing non-Markovian dynamics, and develop a mean-field description via coupled delay differential equations for $m(t)$ and the latent fractions. They show that latency induces deterministic oscillations in the average opinion with a frequency controlled by the latency $l$, and that consensus arises only as a finite-size effect; heterogeneous latencies can damp the oscillations, and an alternative latency rule can eliminate them. The work provides a mechanism for spontaneous opinion swings seen in real social systems and extends latent-voter models to non-Markovian dynamics, with potential applications to elections and other cyclic social processes.

Abstract

The cognitive process of opinion formation is often characterized by stubbornness or resistance of agents to changes of opinion. To capture such a feature we introduce a constant latency time in the standard voter model of opinion dynamics: after switching opinion, an agent must keep it for a while. This seemingly simple modification drastically changes the stochastic diffusive behavior of the original model, leading to deterministic dynamical oscillations in the average opinion of the agents. We explain the origin of the oscillations and develop a mathematical formulation of the dynamics that is confirmed by extensive numerical simulations. We further characterize the rich phase space of the model and its asymptotic behavior. Our work offers insights into understanding and modeling opinion swings in diverse social contexts.

Figures (9)

• Figure 1: (a) Results of US presidential elections in a sample of swing statesDVN/42MVDX_2017. The dynamic shows oscillating behavior with a period set by the occurrence of elections every 4 years. (b) Evolution of the magnetization $m(t)$ for a single realization of the ordinary voter model and of the voter model with latency, for a population of $N=1000$ agents. Note how the original model can be recovered as a special case of the LVM by setting a latency time $l=0$. (c) Simulation results and numerical solution of eq. \ref{['eq_lam']} for the fraction of agents in the latency state $\lambda(t)$, with latency time $l=7$ and $N=1000$ agents (d) The numerical solution with $l=5$ for $m(t)$ and $\Delta \phi(t)$ shows that these quantities oscillate with a quarter-period shift (markers on peaks of $m$ correspond to zeros of $\Delta \phi$).
• Figure 2: (a) PSD of the main frequency of $m$ (computed with the Fast Fourier Transform method) as a function of $l$, for both numerical solution and model simulations (fluctuations are negligible also in the latter case). (b) Period of $m(t)$ (computed as the average distance between peaks) as a function of $l$, for both numerical solution and simulations. (c) Minimum distance from the consensus state, $1-\max_{s\le t}|m(s)|$, achieved by the numerical solution as a function of $N$. (d) Time to reach the consensus state in simulations as a function of the population size $N$.
• Figure S1: Oscillations in elections results for all US states (except DC). $+1$ represents a vote for Democrats and $-1$ for Republicans (votes for third parties are discarded). The so-called swing states are those that oscillate around $m=0$ (and so they are crucial in determining the global outcome), yet all states in fact swing, following a common trend.
• Figure S2: Full plot of variables with $l=7$ (a) and $l=15$ (b).
• Figure S3: Simulation and approximate solution for $\lambda(t)$ have good overlap for a significant amount of time.