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Complexity
Volume 2017 (2017), Article ID 3250301, 12 pages
https://doi.org/10.1155/2017/3250301
Research Article

On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy

1Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965 Poznań, Poland
2Department of Computational Intelligence, ENGINE-The European Centre for Data Science, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Correspondence should be addressed to Mikołaj Morzy; lp.nanzop.tup@yzrom.jalokim

Received 6 April 2017; Revised 27 July 2017; Accepted 13 August 2017; Published 1 November 2017

Academic Editor: Pasquale De Meo

Copyright © 2017 Mikołaj Morzy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

One of the most popular methods of estimating the complexity of networks is to measure the entropy of network invariants, such as adjacency matrices or degree sequences. Unfortunately, entropy and all entropy-based information-theoretic measures have several vulnerabilities. These measures neither are independent of a particular representation of the network nor can capture the properties of the generative process, which produces the network. Instead, we advocate the use of the algorithmic entropy as the basis for complexity definition for networks. Algorithmic entropy (also known as Kolmogorov complexity or -complexity for short) evaluates the complexity of the description required for a lossless recreation of the network. This measure is not affected by a particular choice of network features and it does not depend on the method of network representation. We perform experiments on Shannon entropy and -complexity for gradually evolving networks. The results of these experiments point to -complexity as the more robust and reliable measure of network complexity. The original contribution of the paper includes the introduction of several new entropy-deceiving networks and the empirical comparison of entropy and -complexity as fundamental quantities for constructing complexity measures for networks.