Random Node-Asynchronous Graph Computations: Novel Opportunities for Discrete-Time State-Space Recursions

For the analysis of data defined over irregular domains, the recent area of graph signal processing extends classical signal processing techniques to the case of graphs. A prominent idea in this context is the notion of graph shift, which provides a foundation for distributed signal processing over networks. Despite its distributed nature, successive implementations of graph shift require a synchronization over the network, which can be a restriction in practical applications. The main goal of this article is to review the recent advancements in random and asynchronous computations over graphs with a specific emphasis on asynchronous graph-filtering applications. This article describes the behavior of the randomized asynchronous graph shift from a linear system theory viewpoint while making connections with asynchronous fixed-point iterations and Markov jumpswitching systems. In the context of graph signal processing, two specific applications are considered. The first one combines polynomial filtering and asynchronous updates to compute the Fiedler vector for spectral clustering. The second one considers the asynchronous implementation of a given rational graph filter. Asynchronous graph filtering is also extended to implement polynomial graph filter banks and graph neural networks in an asynchronous manner. The considered applications have proven mean-squared convergence guarantees.