A common practice in the estimation of the complexity of objects, in particular of graphs, is 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 the object, such 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 not only is it necessary to pre-select a feature of interest where there is one when applying a computable measure such as Shannon Entropy, and to make an arbitrary selection where there is not, but that more general properties, such as the causal likeliness 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.
['Hector Zenil', 'Narsis A. Kiani']
Low Algorithmic Complexity Entropy-deceiving Graphs