Towards the Heider balance -- a cellular automaton with a global neighborhood

TL;DR

This paper investigates a deterministic cellular automaton model for social network balance, revealing how system size influences the likelihood of reaching balanced states and the emergence of limit cycles.

Contribution

It introduces a new synchronous update algorithm for network balance and analyzes the role of symmetries and system size in dynamics and attractor states.

Findings

Probability of reaching balanced states increases with system size

Jammed states become less frequent as N grows

Limit cycles of various lengths are observed

Abstract

We study a simple deterministic map that leads a fully connected network to Heider balance. The map is realized by an algorithm that updates all links synchronously in a way depending on the state of the entire network. We observe that the probability of reaching a balanced state increases with the system size N. Jammed states become less frequent for larger N. The algorithm generates also limit cycles, mostly of length $2$, but also of length 3, 4, 6, 12 or 14. We give a simple argument to estimate the mean size of basins of attraction of balanced states and discuss symmetries of the system including the automorphism group as well as gauge invariance of triad configurations. We argue that both the symmetries play an essential role in the occurrence of cycles observed in the synchronous dynamics realized by the algorithm.