Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks
This work provides theoretical insights into GNNs for researchers in machine learning and graph-based applications, but it is incremental as it builds on existing graph signal processing concepts.
The paper characterizes the representation space of graph neural networks (GNNs) using graph signal processing, showing that architectures with graph convolutional filters have permutation equivariance and stability to topology changes, which explains their scalability and transferability.
Network data can be conveniently modeled as a graph signal, where data values are assigned to nodes of a graph that describes the underlying network topology. Successful learning from network data is built upon methods that effectively exploit this graph structure. In this work, we leverage graph signal processing to characterize the representation space of graph neural networks (GNNs). We discuss the role of graph convolutional filters in GNNs and show that any architecture built with such filters has the fundamental properties of permutation equivariance and stability to changes in the topology. These two properties offer insight about the workings of GNNs and help explain their scalability and transferability properties which, coupled with their local and distributed nature, make GNNs powerful tools for learning in physical networks. We also introduce GNN extensions using edge-varying and autoregressive moving average graph filters and discuss their properties. Finally, we study the use of GNNs in recommender systems and learning decentralized controllers for robot swarms.