Fluctuations and Correlations in a Model of Extreme Introverts and Extroverts

TL;DR

This paper investigates the fluctuations and correlations in a social network model of introverts and extroverts exhibiting the Thouless effect, combining simulations and mean field theory to understand its anomalous behavior.

Contribution

It introduces an analytic mean field approach to study the extreme Thouless effect in the XIE model, revealing identities between correlations and fluctuations and explaining anomalous behaviors.

Findings

Confirmed identities between correlations and fluctuations through simulations

Identified unusual fluctuation patterns similar to the 2D Ising model

Developed a mean field theory that explains anomalous fluctuations

Abstract

Unlike typical phase transitions of first and second order, a system displaying the Thouless effect exhibits characteristics of both at the critical point (jumps in the order parameter and anomalously large fluctuations). An $extreme$ Thousless effect was observed in a recently introduced model of social networks consisting of `introverts and extroverts' ($XIE$). We study the fluctuations and correlations of this system using both Monte Carlo simulations and analytic methods based on a self-consistent mean field theory. Due to the symmetries in the model, we derive identities between all independent two point correlations and fluctuations in three quantities (degrees of individuals and the total number of links between the two subgroups) in the stationary state. As simulations confirm these identities, we study only the fluctuations in detail. Though qualitatively similar to those in the 2D Ising model, there are several unusual aspects, due to the extreme Thouless effect. All these anomalous fluctuations can be quantitatively understood with our theory, despite the mean-field aspects in the approximations. In our theory, we frequently encounter the `finite Poisson distribution' (i.e., $x^{n}/n!$ for $n\in \left[ 0,N\right] $ and zero otherwise). Since its properties appear to be quite obscure, we include an Appendix on the details and the related `finite exponential series' $\sum_{0}^{N}x^{n}/n!$. Some simulation studies of joint degree distributions, which provide a different perspective on correlations, have also been carried out.