KAUST DepartmentBiological and Environmental Sciences and Engineering (BESE) Division
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
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AbstractIn estimating the complexity of objects, in particular, of graphs, it is common practice to rely on graphand 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 see that when applying a computable measure such as the Shannon entropy, not only is it necessary to preselect a feature of interest where there is one, and to make an arbitrary selection where there is not, but also 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 the entropy and entropy rate. We introduce recursive and nonrecursive (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.
CitationZenil H, Kiani NA, Tegnér J (2017) Low-algorithmic-complexity entropy-deceiving graphs. Physical Review E 96. Available: http://dx.doi.org/10.1103/PhysRevE.96.012308.
SponsorsN.A.K. was supported by aVinnovaVINNMERfellowship, Stratneuro. H. Z. was supported by the Swedish Research Council (VR).
PublisherAmerican Physical Society (APS)
JournalPhysical Review E
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