  The area of Data Analytics on graphs promises a paradigm shift as we approach
information processing of classes of data, which are typically acquired on
irregular but structured domains (social networks, various ad-hoc sensor
networks). Yet, despite its long history, current approaches mostly focus on
the optimization of graphs themselves, rather than on directly inferring
learning strategies, such as detection, estimation, statistical and
probabilistic inference, clustering and separation from signals and data
acquired on graphs. To fill this void, we first revisit graph topologies from a
Data Analytics point of view, and 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.
Next, to illustrate 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, a framework for vertex
clustering and graph segmentation is established based on graph spectral
representation (eigenanalysis) which illustrates the power of graphs in various
data association tasks. 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.
