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Many systems comprising entities in interactions can be represented asgraphs, whose structure gives significant insights about how these systemswork. Network theory has undergone further developments, in particular inrelation to detection of communities in graphs, to catch this structure.Recently, an approach has been proposed to transform a graph into a collectionof signals: Using a multidimensional scaling technique on a distance matrixrepresenting relations between vertices of the graph, points in a Euclideanspace are obtained and interpreted as signals, indexed by the vertices. In thisarticle, we propose several extensions to this approach, developing a frameworkto study graph structures using signal processing tools. We first extend thecurrent methodology, enabling us to highlight connections between properties ofsignals and graph structures, such as communities, regularity or randomness, aswell as combinations of those. A robust inverse transformation method is nextdescribed, taking into account possible changes in the signals compared tooriginal ones. This technique uses, in addition to the relationships betweenthe points in the Euclidean space, the energy of each signal, coding thedifferent scales of the graph structure. These contributions open up newperspectives in the study of graphs, by enabling processing of graphs throughthe processing of the corresponding collection of signals, using reliable toolsfrom signal processing. A technique of denoising of a graph by filtering of thecorresponding signals is then described, suggesting considerable potential ofthe approach.

From graphs to signals and back: Identification of network structures using spectral analysis