Emergent bipartiteness in a society of knights and knaves

TL;DR

This paper introduces a social network model inspired by knights-and-knaves puzzles, demonstrating how bipartite structures emerge under certain class proportions and analyzing the phase transition behavior.

Contribution

The study presents a novel network formation model based on social puzzles, revealing conditions for bipartiteness and its phase transition characteristics.

Findings

Networks become bipartite within specific class ratios.

Bipartiteness disappears beyond a threshold class imbalance.

Model exhibits behavior similar to first-order phase transitions.

Abstract

We propose a simple model of a social network based on so-called knights-and-knaves puzzles. The model describes the formation of networks between two classes of agents where links are formed by agents introducing their neighbours to others of their own class. We show that if the proportion of knights and knaves is within a certain range, the network self-organizes to a perfectly bipartite state. However, if the excess of one of the two classes is greater than a threshold value, bipartiteness is not observed. We offer a detailed theoretical analysis for the behaviour of the model, investigate its behaviou r in the thermodynamic limit, and argue that it provides a simple example of a topology-driven model whose behaviour is strongly reminiscent of a first-order phase transitions far from equilibrium.