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

Abstract

Many modern data analytics applications on graphs operate on domains where graph topology is not known a priori, and hence its determination becomes part of the problem definition, rather than serving as prior knowledge which aids the problem solution. Part III of this monograph starts by addressing ways to learn graph topology, from the case where the physics of the problem already suggest a possible topology, through to most general cases where the graph topology is learned from the data. A particular emphasis is on graph topology definition based on the correlation and precision matrices of the observed data, combined with additional prior knowledge and structural conditions, such as the smoothness or sparsity of graph connections. For learning sparse graphs (with small number of edges), the least absolute shrinkage and selection operator, known as LASSO is employed, along with its graph specific variant, graphical LASSO. For completeness, both variants of LASSO are derived in an intuitive way, and explained. An in-depth elaboration of the graph topology learning paradigm is provided through several examples on physically well defined graphs, such as electric circuits, linear heat transfer, social and computer networks, and spring-mass systems. As many graph neural networks (GNN) and convolutional graph networks (GCN) are emerging, we have also reviewed the main trends in GNNs and GCNs, from the perspective of graph signal filtering. We have in particular studied the diffusion process over graphs and have shown that the trend of various improvements on GCNs can also be understood from the graph diffusion perspective. Given that the existing GCNs have been introduced largely in a heuristic manner, the definition of different diffusion processes can also serve as a basis for a new design of GCNs. Tensor representation of lattice-structured graphs is next considered, and it is shown that tensors (multidimensional data arrays) are a special class of graph signals, whereby the graph vertices reside on a high-dimensional regular lattice structure. This part of monograph concludes with two emerging applications in financial data processing and underground transportation networks modeling. By means of portfolio cuts of an asset graph, we show how domain knowledge can be meaningfully incorporated into investment analysis. In the underground traffic example, we demonstrate how graph theory can be used to identify the stations in the London underground network which have the greatest influence on the functionality of the traffic, and proceed, in an innovative way, to assess the impact of a station closure on service levels across the city. Contents 1 Introduction 2 2 Geometrically Defined Graph Topologies 3 3 Graph Topology Based on Signal Similarity 4 4 Learning of Graph Laplacian from Data 8 4.1 Imposing Sparsity on the Connection Metric 10 4.2 Smoothness Constrained Learning of Graph Laplacian .

Where and How to Improve Graph-based Spatio-temporal Predictors

“…Spatio-temporal predictive models fit the data by leveraging on the inductive bias associated with relational information coming from time and space of the data observations [Bai et al, 2020, Pal et al, 2021, Ruiz et al, 2020, Seo et al, 2018, Wu et al, 2019. The spatial domain is typically represented as a graph associated with, e.g., a pixel grid, a 3d mesh, a road map, or a brain network [Li et al, 2018, Shuman et al, 2013, Stankovic et al, 2020. That said, however, the term spatial has to be intended in broader terms referring to any type of functional dependence existing among sensors, well beyond those correlations coming from their physical position.…”

Geometric Deep Lean Learning: Evaluation Using a Twitter Social Network

“…The major challenge in this area is to find a way to depict or encode the structure of graphs so that it can be easily exploited by machine learning models. Within this field, geometric deep learning is an emerging technique to generalise deep learning models to non-Euclidean domains such as certain graphs and manifolds [18][19][20][21][22][23], and has been previously used in graph-wise classification [24], signal processing [25], vertex-wise classification [26], or graph dynamics classification [18].…”