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

The structure of networks describing interactions between entities gives significant insights about how these systems work. Recently, an approach has been proposed to transform a graph into a collection of signals, using a multidimensional scaling technique on a distance matrix representing relations between vertices of the graph as points in a Euclidean space: coordinates are interpreted as components, or signals, indexed by the vertices. In this article, we propose several extensions to this approach: We first extend the current methodology, enabling us to highlight connections between properties of the collection of signals and graph structures, such as communities, regularity or randomness, as well as combinations of those. A robust inverse transformation method is next described, taking into account possible changes in the signals compared to original ones. This technique uses, in addition to the relationships between the points in the Euclidean space, the energy of each signal, coding the different scales of the graph structure. These contributions open up new perspectives by enabling processing of graphs through the processing of the corresponding collection of signals. A technique of denoising of a graph by filtering of the corresponding signals is then described, suggesting considerable potential of the approach.