Perfect cycles in the synchronous Heider dynamics in complete network

TL;DR

This paper analyzes the dynamics of a cellular automaton simulating Heider balance in a complete network, identifying a class of symmetric limit cycles called perfect cycles that preserve energy spectrum and exhibit high symmetry.

Contribution

It introduces the concept of perfect cycles in Heider dynamics, characterizing their symmetry and energy preservation in complete networks.

Findings

Perfect cycles are symmetric and energy-preserving limit cycles.

The symmetry of perfect cycles is linked to network permutation symmetry.

The states in perfect cycles form highly symmetric trajectories in configuration space.

Abstract

We discuss a cellular automaton simulating the process of reaching Heider balance in a fully connected network. The dynamics of the automaton is defined by a deterministic, synchronous and global update rule. The dynamics has a very rich spectrum of attractors including fixed points and limit cycles, the length and number of which change with the size of the system. In this paper we concentrate on a class of limit cycles that preserve energy spectrum of the consecutive states. We call such limit cycles perfect. Consecutive states in a perfect cycle are separated from each other by the same Hamming distance. Also the Hamming distance between any two states separated by $k$ steps in a perfect cycle is the same for all such pairs of states. The states of a perfect cycle form a very symmetric trajectory in the configuration space. We argue that the symmetry of the trajectories is rooted in the permutation symmetry of vertices of the network and a local symmetry of a certain energy function measuring the level of balance/frustration of triads.