  Real-world networks typically exhibit several aspects, or layers, of
interactions among their nodes. By permuting the role of the nodes and the
layers, we establish a new criterion to construct the dual of a network. This
approach allows to examine information from either a node-centric or
layer-centric viewpoint. Through rigorous analytical methods and extensive
simulations, we demonstrate that nodewise and layerwise connectivity measure
different but related aspects of the same system. Leveraging node-layer duality
provides complementary insights, enabling a deeper comprehension of diverse
networks across social science, technology and biology. Taken together, these
findings reveal previously unappreciated features of complex systems and
provide a fresh tool for delving into their structure and dynamics.
