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We show that numerical approximations of Kolmogorov complexity (K) applied tograph adjacency matrices capture some group-theoretic and topologicalproperties of graphs and empirical networks ranging from metabolic to socialnetworks. That K and the size of the group of automorphisms of a graph arecorrelated opens up interesting connections to problems in computationalgeometry, and thus connects several measures and concepts from complexityscience. We show that approximations of K characterise synthetic and naturalnetworks by their generating mechanisms, assigning lower algorithmic randomnessto complex network models (Watts-Strogatz and Barabasi-Albert networks) andhigh Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive theseresults via two different Kolmogorov complexity approximation methods appliedto the adjacency matrices of the graphs and networks. The methods used are thetraditional lossless compression approach to Kolmogorov complexity, and anormalised version of a Block Decomposition Method (BDM) measure, based onalgorithmic probability theory.

Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks