Topological duality in complex networks*

Duality enhances our understanding of complex systems by establishing symmetry properties and multiplicity with important consequences in theoretical studies and real-world applications. However, whether complex networks exhibit a duality is still largely unknown. That is because in the classical formalism, where the network interacting unit is the node, it is hard to define a dual dimension and its relation with the primal counterpart. The recently introduced multilayer network formalism offers a unique opportunity to overcome this limitation. In a multilayer network, nodes get connected within and between different layers, the latter representing different types of connectivity or scales. Here, the basic interacting unit is the node-layer duplet, which allows to naturally define a complex networks' duality. On one side, there is the standard primal nodewise dimension where connectivity is regarded from the nodes' perspective. On the other side, there is the dual layerwise dimension where connectivity is observed from the layers' viewpoint. Through rigorous analytical methods and extensive simulations, we demonstrated that nodewise and layerwise connectivity characterize different-but-related aspects of the same system. Critically, both are essential for fully describing the basic structural properties, but only one is in general better positioned to capture the underlying network (re)organization. Such duality enabled a better understanding and characterization of various real-world networks, spanning social, infrastructure, and biological systems. Notably, we discovered that neurodegeneration in Alzheimer's disease is mostly characterized by the disruption of information transfer between different frequencies of brain activity (dual dimension), rather than between spatially distributed brain areas (primal dimension). By shedding light on previously unappreciated hidden properties, we provide a foundation for future investigations into the structure and dynamics of interconnected systems, with broader implications in a wide range of disciplines, including network science, systems biology, and social network analysis.