Learning Graph Embeddings on Constant-Curvature Manifolds for Change Detection in Graph Streams
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The space of graphs is characterized by a non-trivial geometry, which often complicates performing inference in practical applications. A common approach is to use embedding techniques to represent graphs as points in a conventional Euclidean space, but non-Euclidean spaces are often better suited for embedding graphs. Among these, constant curvature manifolds (CCMs), like hyperspheres and hyperboloids, offer a computationally tractable way to compute metric, yet non-Euclidean, geodesic distances. In this paper, we introduce a novel adversarial graph embedding technique to represent graphs on CCMs, and exploit such a mapping for detecting changes in stationarity in a graph-generating process. To this end, we introduce a novel family of change detection tests operating by means of distances on CCMs. We perform experiments on synthetic graph streams, and on sequences of functional networks extracted from iEEG data with the aim of detecting the onset of epileptic seizures. We show that our methods are able to detect extremely small changes in the graph-generating process, consistently outperforming solutions based on Euclidean embeddings. The general nature of our framework highlights its potential to be extended to other applications characterized by graph data or non-Euclidean geometries.
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