Online Learning Of Expanding Graphs

TL;DR

This paper introduces a novel online algorithm for inferring expanding network topologies from spatiotemporal signals, addressing challenges of growing graphs and real-time updates in dynamic environments.

Contribution

It presents the first online method for expanding graphs using projected proximal gradient descent with recursive covariance updates and node-specific strategies.

Findings

Effective in real-world epidemic and financial networks

Handles increasing graph size efficiently

Analyzes computational complexity and regret

Abstract

This paper addresses the problem of online network topology inference for expanding graphs from a stream of spatiotemporal signals. Online algorithms for dynamic graph learning are crucial in delay-sensitive applications or when changes in topology occur rapidly. While existing works focus on inferring the connectivity within a fixed set of nodes, in practice, the graph can grow as new nodes join the network. This poses additional challenges like modeling temporal dynamics involving signals and graphs of different sizes. This growth also increases the computational complexity of the learning process, which may become prohibitive. To the best of our knowledge, this is the first work to tackle this setting. We propose a general online algorithm based on projected proximal gradient descent that accounts for the increasing graph size at each iteration. Recursively updating the sample covariance matrix is a key aspect of our approach. We introduce a strategy that enables different types of updates for nodes that just joined the network and for previously existing nodes. To provide further insights into the proposed method, we specialize it in Gaussian Markov random field settings, where we analyze the computational complexity and characterize the dynamic cumulative regret. Finally, we demonstrate the effectiveness of the proposed approach using both controlled experiments and real-world datasets from epidemic and financial networks.