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Abstract-Graph representations offer powerful and intuitive ways to describe data in a multitude of application domains. Here, we consider stochastic processes generating graphs and propose a methodology for detecting changes in stationarity of such processes. The methodology is general and considers a process generating attributed graphs with a variable number of vertices/edges, without the need to assume a one-to-one correspondence between vertices at different time steps. The methodology acts by embedding every graph of the stream into a vector domain, where a conventional multivariate change detection procedure can be easily applied. We ground the soundness of our proposal by proving several theoretical results. In addition, we provide a specific implementation of the methodology and evaluate its effectiveness on several detection problems involving attributed graphs representing biological molecules and drawings. Experimental results are contrasted with respect to suitable baseline methods, demonstrating the effectiveness of our approach.

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The space of graphs is often characterised by a nontrivial geometry, which complicates learning and 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 have often been shown to be better suited for embedding graphs. Among these, constant-curvature Riemannian manifolds (CCMs) offer embedding spaces suitable for studying the statistical properties of a graph distribution, as they provide ways to easily compute metric geodesic distances. In this paper, we focus on the problem of detecting changes in stationarity in a stream of attributed graphs. To this end, we introduce a novel change detection framework based on neural networks and CCMs, that takes into account the non-Euclidean nature of graphs. Our contribution in this work is twofold. First, via a novel approach based on adversarial learning, we compute graph embeddings by training an autoencoder to represent graphs on CCMs. Second, we introduce two novel change detection tests operating on CCMs.We perform experiments on synthetic data, as well as two realworld application scenarios: the detection of epileptic seizures using functional connectivity brain networks, and the detection of hostility between two subjects, using human skeletal graphs. Results show that the proposed methods are able to detect even small changes in a graph-generating process, consistently outperforming approaches based on Euclidean embeddings.

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