Bruno Messias Farias de Resende

Bruno Messias Farias de Resende

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.

Bruno Messias F. de Resende, Eric K. Tokuda, Luciano da Fontoura Costa

In the big-data age, tabular data are being generated and analyzed everywhere. As a consequence, finding and understanding the relationships between the features in these data are of great relevance. Here, to encompass these relationships, we propose a graph-based method that allows individual, group and multi-scale analyses. The method starts by mapping the tabular data into a weighted directed graph using the Shapley additive explanations technique. With this graph of relationships, we show that the inference of the hierarchical modular structure obtained by the Nested Stochastic Block Model (nSBM) as well as the study of the spectral space of the magnetic Laplacian can help us identify the classes of features and unravel non-trivial relationships. As a case study, we analyzed a socioeconomic survey conducted with students in Brazil: the PeNSE survey. The spectral embedding of the columns suggested that questions related to physical activities form a separate group. The application of the nSBM approach not only corroborated with that but allowed complementary findings about the modular structure: some groups of questions showed a high adherence with the divisions qualitatively defined by the designers of the survey. As opposed to the structure obtained by the spectrum, questions from the class Safety were partly grouped by our method in the class Drugs. Surprisingly, by inspecting these questions, we observed that they were related to both these topics, suggesting an alternative interpretation of these questions. These results show how our method can provide guidance for tabular data analysis as well as the design of future surveys.