To address the peculiarities of directed and/or signed graphs, new Laplacian
operators have emerged. For instance, in the case of directionality, we
encounter the magnetic operator, dilation (which is underexplored), operators
based on random walks, and so forth. The definition of these new operators
leads to the need for new studies and concepts, and consequently, the
development of new computational tools. But is this really necessary? In this
work, we define the concept of effective adjacency matrices that arise from the
definition of deformed Laplacian operators such as magnetic, dilation, and
signal. These effective matrices allow mapping generic graphs to a family of
unsigned, undirected graphs, enabling the application of the well-explored
toolkit of measures, machine learning methods, and renormalization groups of
undirected graphs. To explore the interplay between deformed operators and
effective matrices, we show how the Hodge-Helmholtz decomposition can assist us
in navigating this complexity.

通过定义磁场、扩张、信号等变形拉普拉斯算子所产生的有效邻接矩阵，我们将通用图映射到一个无向无符号图族，从而能够应用其工具集来进行度量、机器学习和归一化等操作。同时，我们探索了变形算子和有效矩阵之间的相互作用，并展示了如何利用 Hodge-Helmholtz 分解帮助我们在复杂度中进行导航。