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 outputs of 1-dimensional and 2-dimensional Turing machines to determine the complexity values of binary strings and 2-dimensional binary matrices 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 has 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 sub-graphs within a graph using the entire symmetric group of its vertices permutation and via unique permutations we call automorphic subsets, which is a special subset of the symmetric group. We also analyze if edges will be grouped closer to their respective sub-graphs in terms of the average algorithmic information contribution. This analysis has been done in order to ascertain if Algorithmic Information Theory can be a viable theory in understanding substructures within graphs and ultimately as a foundation to create frameworks of measuring and analyzing complexity.

翻译：通过不同复杂系统间的相互作用来理解自然现象已成为科学探究日益关注的焦点。定义复杂性并实际测量其值是一个持续争论的话题，目前尚未建立既理论严谨又计算实用的标准框架。当前，算法信息论领域正尝试对复杂性进行形式化定义。该领域通过研究一维和二维图灵机的输出，分别确定二进制字符串和二维二进制矩阵的复杂度值，已取得显著进展。利用这些复杂度值，Zenil等人于2018年提出的块分解方法被用于近似计算图的邻接矩阵复杂度，该方法在基于复杂度值进行图分组方面取得了相对成功。我们采用该方法结合另一种称为边扰动的方法，通过顶点置换的完整对称群以及我们称为自同构子集（对称群的特殊子集）的独特置换，穷举性地判定图中是否存在连接两个子图的边。我们还分析了边是否会根据平均算法信息贡献度更紧密地聚集到其各自所属的子图。进行此项分析旨在确认算法信息论能否成为理解图内子结构的有效理论，并最终为建立复杂性测量与分析框架奠定基础。