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

TL;DR

This paper demonstrates that Kolmogorov complexity approximations of graph adjacency matrices reveal correlations with automorphism group sizes and topological features, distinguishing between different network models and natural networks.

Contribution

It introduces methods to approximate Kolmogorov complexity for graphs, linking complexity measures to graph symmetries and topological properties, with applications to real-world networks.

Findings

Kolmogorov complexity correlates with automorphism group size.

Complex network models have lower complexity than random graphs.

Approximation methods effectively characterize network generating mechanisms.

Abstract

We show that numerical approximations of Kolmogorov complexity (K) applied to graph adjacency matrices capture some group-theoretic and topological properties of graphs and empirical networks ranging from metabolic to social networks. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We show that approximations of K characterise synthetic and natural networks by their generating mechanisms, assigning lower algorithmic randomness to complex network models (Watts-Strogatz and Barabasi-Albert networks) and high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) measure, based on algorithmic probability theory.