Signal Processing over Dynamic Graphs

Abstract

Extending the concepts of classical signal processing to graphs, a wide array of methods have come to the fore, including filtering, reconstruction, classification, and sampling. Existing approaches in graph signal processing consider a known and static topology, i.e., fixed number of nodes and a fixed edge support. Two types of tasks stand out, namely, topology inference, where the edge support along with their weights are estimated from signals; and data processing, where existing data and the known topology are used to perform different tasks. However, such tasks become quite challenging when the network size and support changes over time. Particularly, these challenges involve adapting to the changing topology, data distributions and dealing with unknown topological information. The latter manifests for example, when new nodes are available to attach to the graph but their connectivity is uncertain as is the case in cold start graph-based recommender systems.
The key contribution of this dissertation is proposing methodologies for signal processing over dynamic networks which are aimed at the two aforementioned tasks. For dynamic networks with incoming nodes, we process signals by introducing a parametric stochastic attachment model. In this model, the incoming nodes connect with probability to existing nodes with certain weights. This uncertainty allows us to model input output relations and allows us to cast them in the context of different graph signal processing tasks. We learn the model attachment parameters in a task-aware setting, allowing us to interpret topology identification in task-aware settings. Separately, we also propose filter design strategies for processing signals both at the incoming and existing nodes using stochastic attachment models.
Another contribution of this dissertation is to extend graph signal processing with graph filters to the scenario where the graph keeps growing in size with streaming data. We propose online graph filter design which updates the filter online, based on incoming nodes. We design this both for scenarios where the incoming node connectivity is known and unknown. In the unknown connectivity case, we study the performance difference between knowing and not knowing the topology and how the stochastic attachment influences it. We also show that by adapting the stochastic attachment, we can learn faster from the data stream.
Finally,we consider the task of topology decomposition and identification for dynamic networks with fixed nodes but changing edge support. We build a tensor of partially observed adjacency matrices corresponding to such a dynamic topology and express this in terms of underlying latent graphs and their temporal signatures. Furthermore, we account for the time-varying graph signals as a prior to aid identifying these latent graphs and missing components of the topology. These latent graphs are individually and collectively expressive and provide interpretable decompositions along with outperforming traditional structure agnostic low-rank decompositions.