Modeling interacting dynamic networks: III. Extraordinary properties in a population of extreme introverts and extroverts

TL;DR

This paper studies a simplified model of social networks with extreme introverts and extroverts, revealing an extraordinary transition and extreme Thouless effect, with analytical solutions for the steady-state distribution.

Contribution

It introduces the XIE model, a minimal system showing extreme properties, and provides analytical solutions for its steady-state behavior.

Findings

Fraction of I-E links jumps sharply at N_I=N_E

Distribution of links is flat at N_I=N_E, indicating a Thouless effect

Analytical degree distributions match simulations away from critical point

Abstract

Recently, we introduced dynamic networks with preferred degrees, showing that interesting properties are present in a single, homogeneous system as well as one with two interacting networks. While simulations are readily performed, analytic studies are challenging, due mainly to the lack of detailed balance in the dynamics. Here, we consider the two-community case in a special limit: a system of extreme introverts and extroverts - the XIE model. Surprising phenomena appear, even in this minimal model, where the only control parameters are the numbers of each subgroup: $N_{I,E}$. Specifically, an extraordinary transition emerges when $N_I$ crosses $N_E$. For example, the fraction of total number of I-E links jumps from $\thicksim 0$ to $\thicksim 1$. In a $N_I=N_E$ system, this fraction performs a pure random walk so that its distribution displays a flat plateau across most of $[0,1]$, with the edges vanishing as $(N_{I,E})^{-0.38}$ for large systems. Thus, we believe the XIE model exhibits an extreme Thouless effect. For this limiting model, we show that detailed balance is restored and explicitly find the microscopic steady-state distribution. We then use a mean-field approach to find analytic expressions for the degree distributions that are in reasonably good agreement with simulations, provided $N_I$ is not too close to $N_{E}$.