Grant: Discovery Grants Program - Individual

Network analysis has become an essential part of data science, because data is frequently presented in networked form. Links can represent data that are physically or virtually linked; links could also indicate that data are similar or correlated. A robust approach to network analysis should not depend on one particular network. Namely, in many cases, networks change over time, or the information on which the graph is based contains errors. Instead, network analysis should be founded in a common framework that applies to collections of similar graphs. Network tasks such as community finding and link prediction operate on an assumption that link formation is guided by hidden features of the nodes. Graphons provide a versatile way to model this latent feature space. Graphons provide a 'landscape' of link probabilities. To sample a graph from a graphon, nodes are randomly sampled from the feature space, and the placement of the nodes determines the probability that nodes are linked. Graphs sampled from a graphon will be similar in structure. Thus graphons can provide a common framework for network analysis. Network analysis in this framework relies on finding the latent feature space and the placement of the vertices therein. This is known as graphon estimation. In the proposed research, we will develop new methods for graphon estimation. The graphons described considered in the proposed research provide nuanced models that go beyond simple clustering of the nodes into communities with similar features. In many applications, measurements are associated with the nodes of a network. These measurements can be interpreted as a signal on the graph. Graph Signal Processing (GSP) has shown to be an effective tool in applications such as network failure assessment, detection of false data attacks in smart grids, and functional analysis of brain signals. Traditionally, GSP is based on a Fourier transform that is derived from the graph. Graphons provide a broader framework. GSP based on graphons has only recently become a topic of study, and our recent work is at the forefront. The proposed research will help establish the foundations of graphon-based GSP.