Heider balance on Archimedean lattices and cliques

TL;DR

This paper explores how the Heider balance function behaves on various planar networks with triangles, revealing that local topology influences the balance and proving that perfect balance cannot be achieved at positive temperatures.

Contribution

It provides a comprehensive analysis of the Heider balance on different planar networks and offers a mathematical proof regarding the impossibility of perfect balance at T>0.

Findings

The shape of U(T) depends on local topology.

Asynchronous updates differ from synchronous in results.

Perfect structural balance is impossible at T>0.

Abstract

We investigate the work function $U(T)$ for the Heider balance, driven by a thermal noise $T$, on several planar networks that contain separated triangles, pairs of triangles, chains of triangles and complex structures of triangles. In simulations, the heat-bath algorithm is applied. Two schemes of link values updating are considered: synchronous and asynchronous (sequential). The latter results are compared with analytical calculations for small cliques. We argue that the actual shape of $U(T)$ is a consequence of a local topology rather than of a macroscopic ordering. Finally, we present the mathematical proof that for any planar lattice, perfect structural (Heider) balance is unreachable at $T>0$.