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.

翻译：通过不同复杂系统间的相互作用来理解自然现象，已成为科学研究日益关注的焦点。如何定义复杂性并实际测量它，目前仍存在持续争论，尚未建立起兼具理论严谨性与计算实用性的标准框架。当前，算法信息论领域正尝试对复杂性进行形式化定义。该领域通过分别使用一维和二维图灵机研究二进制字符串与二维二进制矩阵的复杂度值，已取得重要进展。基于这些复杂度值，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，该方法在根据复杂度值对图进行分组方面取得了较好效果。本研究结合块分解方法与另一种称为边扰动的方法，通过遍历顶点置换的完全对称群以及我们称为自同构子集（对称群的特殊子集）的唯一置换，系统性地判定图中连接两个子图的边是否可被识别。我们还通过平均算法信息贡献度，分析边是否会被归类到更接近其对应子图的组别中。此项分析旨在验证算法信息论能否成为理解图子结构的有效理论，并为复杂性的测量分析框架奠定基础。研究发现：在30个测试图中，有29个图的连接边被成功识别为具有最高平均信息贡献度，其中16个图中该边与次高贡献边的距离大于log_2(2)。此外，在边分组任务中，完全对称群的表现优于自同构子集。