Measuring Complexity through Average Symmetry

TL;DR

This paper introduces a new complexity measure based on average symmetry breaking, which effectively distinguishes between different structural patterns and is applicable to various types of objects and networks.

Contribution

It proposes a novel, easily computable complexity measure that overcomes limitations of existing measures by focusing on symmetry breaking and generalizes to continuous and network structures.

Findings

Consistently discriminates between patterns, randomness, and self-similarity.

Applicable to small and large scale structures.

Easily generalizable to continuous cases and networks.

Abstract

This work introduces a complexity measure which addresses some conflicting issues between existing ones by using a new principle - measuring the average amount of symmetry broken by an object. It attributes low (although different) complexity to either deterministic or random homogeneous densities and higher complexity to the intermediate cases. This new measure is easily computable, breaks the coarse graining paradigm and can be straightforwardly generalised, including to continuous cases and general networks. By applying this measure to a series of objects, it is shown that it can be consistently used for both small scale structures with exact symmetry breaking and large scale patterns, for which, differently from similar measures, it consistently discriminates between repetitive patterns, random configurations and self-similar structures.