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Many modern data analytics applications on graphs operate on domains wheregraph topology is not known a priori, and hence its determination becomes partof the problem definition, rather than serving as prior knowledge which aidsthe problem solution. Part III of this monograph starts by addressing ways tolearn graph topology, from the case where the physics of the problem alreadysuggest a possible topology, through to most general cases where the graphtopology is learned from the data. A particular emphasis is on graph topologydefinition based on the correlation and precision matrices of the observeddata, combined with additional prior knowledge and structural conditions, suchas the smoothness or sparsity of graph connections. For learning sparse graphs(with small number of edges), the least absolute shrinkage and selectionoperator, known as LASSO is employed, along with its graph specific variant,graphical LASSO. For completeness, both variants of LASSO are derived in anintuitive way, and explained. An in-depth elaboration of the graph topologylearning paradigm is provided through several examples on physically welldefined graphs, such as electric circuits, linear heat transfer, social andcomputer networks, and spring-mass systems. As many graph neural networks (GNN)and convolutional graph networks (GCN) are emerging, we have also reviewed themain trends in GNNs and GCNs, from the perspective of graph signal filtering.Tensor representation of lattice-structured graphs is next considered, and itis shown that tensors (multidimensional data arrays) are a special class ofgraph signals, whereby the graph vertices reside on a high-dimensional regularlattice structure. This part of monograph concludes with two emergingapplications in financial data processing and underground transportationnetworks modeling.

Graph Signal Processing -- Part III: Machine Learning on Graphs, from Graph Topology to Applications