Potential grouping of nodes induced by higher-order structures in complex networks

Higher-order clustering patterns in simplicial financial systems

The interest in induced higher-order relational and multidimensional structures embedded in the financial complex dataset is considered within the applied algebraic topology framework. The aim is to transcend the binary correlations when the interactions of the underlying system are stored in the entries of the cross-correlation matrix. By applying different criteria, we examined aggregations of firms through higher-order clustering of the financial system. The outcome is the extraction of patterns that appear in assemblages of firms due to their multidimensional properties embedded in the cross-correlation matrices. Results are compatible with classifying firms into clusters due to the industry they belong to. Furthermore, the novel and mixed collections of firms are revealed based on the applied mathematical approach. In the broader context, results shed light on the higher-order organization of interactions embedded in the cross-correlation matrix and, as a consequence, extract patterns of collective behavior within a complex system.

Dark soliton detection using persistent homology

Classifying images often requires manual identification of qualitative features. Machine learning approaches including convolutional neural networks can achieve accuracy comparable to human classifiers but require extensive data and computational resources to train. We show how a topological data analysis technique, persistent homology, can be used to rapidly and reliably identify qualitative features in experimental image data. The identified features can be used as inputs to simple supervised machine learning models, such as logistic regression models, which are easier to train. As an example, we consider the identification of dark solitons using a dataset of 6257 labeled atomic Bose-Einstein condensate density images.