Inference From Complex Networks: Role of Symmetry and Applicability to Images - Capobianco, Enrico (Frontiers in Applied Mathematics and Statistics, 2020)
Symmetry is a mathematical concept only partially explored in networks, especially at the applicative level. One reason is a certain lack of interpretable inference obtained from networks. While the network systemic associations (links) between entities (nodes) emerge from the underlying dependence structure, this latter is only partially explicit via the established direct interactors and remains to a certain extent latent (distant node predicted paths). Verifiability of significant hubs, connectors, paths, and modules allows to build a knowledge base useful to infer latencies and/or validate complex associations. When symmetry is searched in images, reflection, translation and rotation are applicable transformations in n-dimensional Euclidean space that computational algorithms target. There is symmetry when original and transformed images cannot be distinguished. Once collected together, such transformations form an automorphism group, indicating a stable and robust global characteristic. It is common to step from images to quantifiable features for conducting inference. Deep learning is typically used to classify whole images reconstructed from the myriads of features in which these images are decomposed. However, with images considered at multiple scales and locations, symmetries are valuable for describing local characteristics. Casting local features into a network framework enables their associations to be explored by similarity or dissimilarity criteria. This is quite intriguing because network configurations may display topological features and connectivity patterns associated with synchronization and symmetry that reduce the redundancy of features to more compact functional descriptions. Then, identifying anomalies from unusual events, behaviors, patterns would spot network vulnerabilities and signs of symmetry breaking.