Data Analytics on Graphs Part I: Graphs and Spectra on Graphs

The area of Data Analytics on graphs promises a paradigm shift, as we approach information processing of new classes of data which are typically acquired on irregular but structured domains (such as social networks, various ad-hoc sensor networks). Yet, despite the long history of Graph Theory, current approaches tend to focus on aspects of optimisation of graphs themselves rather than on eliciting strategies relevant to the objective application of the graph paradigm, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. In order to bridge this gap, we first revisit graph topologies from a Data Analytics point of view, to establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Through a number of carefully chosen examples, we demonstrate that the isomorphic nature of graphs enables both the basic properties of data observed on graphs and their descriptors (features) to be preserved throughout the data analytics process, even in the case of reordering of graph vertices, where classical approaches fail. Next, to illustrate the richness and flexibility of estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, benefiting from enhanced degrees of freedom associated with graph representations, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which demonstrates the power of graphs in various data association tasks, from image clustering and segmentation trough to low-dimensional manifold representation. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data.