Opinion Dynamics on Networks under Correlated Disordered External Perturbations

Abstract

We study an influence network of voters subjected to correlated disordered external perturbations, and solve the dynamical equations exactly for fully connected networks. The model has a critical phase transition between disordered unimodal and ordered bimodal distribution states, characterized by an increase in the vote-share variability of the equilibrium distributions. The fluctuations (variance and correlations) in the external perturbations are shown to reduce the impact of the external influence by increasing the critical threshold needed for the bimodal distribution of opinions to appear. The external fluctuations also have the surprising effect of driving voters towards biased opinions. Furthermore, the first and second moments of the external perturbations are shown to affect the first and second moments of the vote-share distribution. This is shown analytically in the mean field limit, and confirmed numerically for fully connected networks and other network topologies. Studying the dynamic response of complex systems to disordered external perturbations could help us understand the dynamics of a wide variety of networked systems, from social networks and financial markets to amorphous magnetic spins and population genetics.

Appendices

Appendix 1: The Dependence of $\sigma _{\nu }^2$ on Total Fluctuations

Consider Eq 21. In the fully symmetric case, $(\bar{N_0}=\bar{N_1}\equiv \bar{N}, \sigma _0=\sigma _1\equiv \sigma$ and $n_0=n_1\equiv n)$. This implies that $\bar{n}_T=N+\bar{N_0}+\bar{N_1}-1=N+2\bar{N}-1$, and Eq. (21) becomes

$$\begin{aligned} n=\left[ \frac{\bar{N}\bar{n}_T^2-(\bar{n}_T-2\bar{N})(\sigma ^2+\text{ cov }_{01})}{\bar{n}_T^2+2(\text{ cov }_{01}+\sigma ^2)}\right] . \end{aligned}$$

(44)

In turn, the variance $\sigma _{\nu }^2$ is given by Eq. (16)

$$\begin{aligned} \sigma _{\nu }^2=\frac{1}{4N}\left( \frac{N}{2n+1}+\frac{2n}{2n+1}\right) . \end{aligned}$$

(45)

Thus, even without the linear approximation, $\sigma _{\nu }^2$ is a function of the total fluctuation $\sigma ^2+\text{ cov }_{01}$.

Appendix 2: Generating the Lognormal Distributions

In this appendix, we describe the methods used to generate the bivariate lognormal distributions for the various simulations.

Consider a semi-symmetric bivariate lognormal distributions ($\bar{N_0}\ne \bar{N_1}$ and $\sigma _0^2=\sigma _1^2\equiv \sigma ^2$) as used in Fig. 3. In this case, we fix the values of $\bar{N_0}, \bar{N_1}, \sigma$ and $\rho _{01}$, and determine the values of $\mu _1$, $\mu _2$, $\sigma _{X_1}$, $\sigma _{X_2}$ and $\rho$ by solving Eqs. (35) and (36). From these Eqs we can extract the values of $\sigma _{X_1}$ and $\mu _1$:

$$\begin{aligned} \sigma _{X_1}^2=\ln \left( \frac{\sigma ^2}{\bar{N_0}^2}+1\right) \end{aligned}$$

(46)

and

$$\begin{aligned} \mu _1=\ln (\bar{N_0})-\frac{\sigma _{X_1}^2}{2}. \end{aligned}$$

(47)

The values of $\sigma _{X_2}$ and $\mu _2$ can be obtained similarly. The covariance $\text{ cov }(X_1, X_2)$ is extracted from Eq. (36):

$$\begin{aligned} \text{ cov }(X_1, X_2)=\ln \left( \frac{\rho _{01}\sigma ^2}{\bar{N_0}\bar{N_1}}+1\right) . \end{aligned}$$

(48)

Finally, the expression for $\rho$ is obtained from $\rho =\text{ cov }(X_1, X_2)/(\sigma _{X_1}\sigma _{X_2})$. Having determined the values of $\mu _1$, $\mu _2$, $\sigma _{X_1}$, $\sigma _{X_2}$ and $\rho$ we can generate a bivariate normal variable $(X_1, X_2)^{T}$, which is then used to generate the external influence vector $\mathcal {N}=e^{\mathbf {X}}$. The fully symmetric bivariate lognormal distributions as the ones used in Figs. 1, 2 and 4, 5, 6 can be obtained by setting $\bar{N_0}=\bar{N_1}\equiv \bar{N}$.