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

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.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical Mechanics and its Application

A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression algorithms fall short at characterizing patterns other than statistical ones not different to entropy estimations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure $m$ calculated from the output distribution of small Turing machines. We introduce and justify finite approximations $m_k$ that have been used in some applications as an alternative to lossless compression algorithms for approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of the relevant properties of both $m$ and $m_k$ and compare them to Levin's Universal Distribution. We provide error estimations of $m_k$ with respect to $m$. Finally, we present an application to integer sequences from the Online Encyclopedia of Integer Sequences which suggests that our AP-based measures may characterize non-statistical patterns, and we report interesting correlations with textual, function and program description lengths of the said sequences.Comment: As accepted by the journal Complexity (Wiley/Hindawi

Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures

We show that real-value approximations of Kolmogorov-Chaitin complexity K(s) using the algorithmic coding theorem, as calculated from the output frequency of a large set of small deterministic Turing machines with up to 5 states (and 2 symbols), is consistent with the number of instructions used by the Turing machines producing s, which in turn is consistent with strict integer-value program-size complexity (based on our knowledge of the smallest machine in terms of the number of instructions used). We also show that neither K(s) nor the number of instructions used manifests any correlation with Bennett's Logical Depth LD(s), other than what's predicted by the theory (shallow and non-random strings have low complexity under both measures). The agreement between the theory and the numerical calculations shows that despite the undecidability of these theoretical measures, the rate of convergence of approximations is stable enough to devise some applications. We announce a Beta version of an Online Algorithmic Complexity Calculator (OACC) implementing these methods

Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

We show that there exists a universal quantum Turing machine (UQTM) that can simulate every other QTM until the other QTM has halted and then halt itself with probability one. This extends work by Bernstein and Vazirani who have shown that there is a UQTM that can simulate every other QTM for an arbitrary, but preassigned number of time steps. As a corollary to this result, we give a rigorous proof that quantum Kolmogorov complexity as defined by Berthiaume et al. is invariant, i.e. depends on the choice of the UQTM only up to an additive constant. Our proof is based on a new mathematical framework for QTMs, including a thorough analysis of their halting behaviour. We introduce the notion of mutually orthogonal halting spaces and show that the information encoded in an input qubit string can always be effectively decomposed into a classical and a quantum part.Comment: 18 pages, 1 figure. The operation R is now really a quantum operation (it was not before); corrected some typos, III.B more readable, Conjecture 3.15 is now a theore