Algorithmic Information Theory for Graph Edge Grouping and Substructure Analysis

TL;DR

This paper explores the application of Algorithmic Information Theory to analyze graph substructures, specifically focusing on edge grouping and complexity measurement, using the Block Decomposition Method and edge perturbation techniques.

Contribution

It introduces a novel approach combining Algorithmic Information Theory with edge perturbation to identify subgraph connections and analyze graph complexity.

Findings

Edges with highest information contribution identified in 29/30 graphs

Symmetric group outperformed automorphic subsets in edge grouping

Edges were often closer to their respective subgraphs in terms of information contribution

Abstract

Understanding natural phenomenon through the interactions of different complex systems has become an increasing focus in scientific inquiry. Defining complexity and actually measuring it is an ongoing debate and no standard framework has been established that is both theoretically sound and computationally practical to use. Currently, one of the fields which attempts to formally define complexity is in the realm of Algorithmic Information Theory. The field has shown advances by studying the complexity values of binary strings and 2-dimensional binary matrices using 1-dimensional and 2-dimensional Turing machines, respectively. Using these complexity values, an algorithm called the Block Decomposition Method developed by Zenil, et al. in 2018, has been created to approximate the complexity of adjacency matrices of graphs which have found relative success in grouping graphs based on their complexity values. We use this method along with another method called edge perturbation to exhaustively determine if an edge can be identified to connect two subgraphs within a graph using the entire symmetric group of its vertices permutation and via unique permutations we call automorphic subsets, which are a special subset of the symmetric group. We also analyze if edges will be grouped closer to their respective subgraphs in terms of the average algorithmic information contribution. This analysis ascertains if Algorithmic Information Theory can serve as a viable theory for understanding graph substructures and as a foundation for frameworks measuring and analyzing complexity. The study found that the connecting edge was successfully identified as having the highest average information contribution in 29 out of 30 graphs, and in 16 of these, the distance to the next edge was greater than log_2(2). Furthermore, the symmetric group outperformed automorphic subsets in edge grouping.