The Many-agent limit of the Extreme Introvert-Extrovert model

TL;DR

This paper analyzes a simplified model of social agents with introvert and extrovert behaviors, deriving exact results for large populations and exploring phase transitions and cooperative dynamics.

Contribution

It provides an exact analysis of the many-agent limit of a social interaction model with long-range couplings and introduces variations with preferential attachment behaviors.

Findings

Exact solution for large N behavior of the model.

Identification of a phase transition related to link density.

Analysis of fluctuations near the phase transition.

Abstract

We consider a toy model of interacting extrovert and introvert agents introduced earlier by Liu et al [Europhys. Lett. {\bf 100} (2012) 66007]. The number of extroverts, and introverts is $N$ each. At each time step, we select an agent at random, and allow her to modify her state. If an extrovert is selected, she adds a link at random to an unconnected introvert. If an introvert is selected, she removes one of her links. The set of $N^2$ links evolves in time, and may be considered as a set of Ising spins on an $N \times N$ square-grid with single-spin-flip dynamics. This dynamics satisfies detailed balance condition, and the probability of different spin configurations in the steady state can be determined exactly. The effective hamiltonian has long-range multi-spin couplings that depend on the row and column sums of spins. If the relative bias of choosing an extrovert over an introvert is varied, this system undergoes a phase transition from a state with very few links to one in which most links are occupied. We show that the behavior of the system can be determined exactly in the limit of large $N$. The behavior of large fluctuations in the total numer of links near the phase transition is determined. We also discuss two variations, called egalitarian and elitist agents, when the agents preferentially add or delete links to their least/ most-connected neighbor. These shows interesting cooperative behavior.