Low Algorithmic Complexity Entropy-deceiving Graphs

TL;DR

This paper demonstrates that computable measures like Shannon Entropy can be misleading when estimating graph complexity, especially for graphs constructed from sequences with specific properties, highlighting their limitations.

Contribution

It introduces uncomputable graph constructions to show how entropy-based measures can be manipulated and misrepresent true graph complexity.

Findings

Entropy values vary significantly for different graph descriptions.

Computable measures can be deceived by specific graph constructions.

Uncomputable graphs reveal the limitations of entropy-based complexity estimates.

Abstract

In estimating the complexity of objects, in particular of graphs, it is common practice to rely on graph- and information-theoretic measures. Here, using integer sequences with properties such as Borel normality, we explain how these measures are not independent of the way in which an object, such as a graph, can be described or observed. From observations that can reconstruct the same graph and are therefore essentially translations of the same description, we will see that when applying a computable measure such as Shannon Entropy, not only is it necessary to pre-select a feature of interest where there is one, and to make an arbitrary selection where there is not, but also that more general properties, such as the causal likelihood of a graph as a measure (opposed to randomness), can be largely misrepresented by computable measures such as Entropy and Entropy rate. We introduce recursive and non-recursive (uncomputable) graphs and graph constructions based on these integer sequences, whose different lossless descriptions have disparate Entropy values, thereby enabling the study and exploration of a measure's range of applications and demonstrating the weaknesses of computable measures of complexity.