The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy

TL;DR

This paper refines the principle of maximum entropy by incorporating the generative mechanisms of systems, using algorithmic complexity to distinguish between different types of randomness, and introduces a new graph algorithm for network analysis.

Contribution

It introduces an algorithmic refinement to classical Maxent, applying it to network graphs with a new MARPA algorithm based on algorithmic randomness.

Findings

Reprogrammability asymmetry is linked to non-monotonic algorithmic probability.

The refined Maxent distinguishes between truly random and pseudo-random structures.

The MARPA algorithm generalizes previous network randomness approaches.

Abstract

The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the rest of the elements in the distribution. Because classical entropy-based Maxent collapses cases confounding all distinct degrees of randomness and pseudo-randomness, here we take into consideration the generative mechanism of the systems considered in the ensemble to separate objects that may comply with the principle under some restriction and whose entropy is maximal but may be generated recursively from those that are actually algorithmically random offering a refinement to classical Maxent. We take advantage of a causal algorithmic calculus to derive a thermodynamic-like result based on how difficult it is to reprogram a computer code. Using the distinction between computable and algorithmic randomness we quantify the cost in information loss associated with reprogramming. To illustrate this we apply the algorithmic refinement to Maxent on graphs and introduce a Maximal Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a generalisation over previous approaches. We discuss practical implications of evaluation of network randomness. Our analysis provides insight in that the reprogrammability asymmetry appears to originate from a non-monotonic relationship to algorithmic probability. Our analysis motivates further analysis of the origin and consequences of the aforementioned asymmetries, reprogrammability, and computation.