Change Detection in Graph Streams by Learning Graph Embeddings on Constant-Curvature Manifolds

Abstract

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.

Adversarial autoencoders with constant-curvature latent manifolds

“…During the regularisation phase, we train the critic network to discriminate between samples coming from the encoder and samples from the true prior P M (θ), and then we update the encoder to fool the critic network. By matching the posterior to P M (θ), the network is implicitly constrained to embed input data on the CCM, and the solution to the adversarial game can be obtained from Equation (1) as: However, this implicit optimisation is often not sufficient for the network to effectively learn the non-Euclidean geometry of the CCM, as also highlighted by the experimental results shown in [5]. Consequently, here we also train the encoder network to maximise the membership degree of the embeddings to M, so that the loss landscape is explicitly modified in favour of those embeddings that lie exactly on the CCM.…”

“…Many works in recent literature have highlighted that non-Euclidean geometry can naturally arise in many application domains, with constant-curvature Riemannian manifolds (CCMs), e.g., hyperspherical and hyperbolic manifolds, playing a prominent role as embedding spaces for a variety of data distributions. Among these, representation learning models for images [1], hierarchical data structures like trees and text [2,3], relational networks [4], brain functional connectivity networks [5], and dissimilarity-based datasets [6] have been shown to benefit from non-Euclidean embedding manifolds. Analysing non-Euclidean data via unsupervised deep learning, in particular, has been object of study by several recent works, with interesting results from both theoretical and practical perspectives [7].…”

“…Among the most recent works studying the latent space of autoencoders we cite [8,9,10], highlighting the non-Euclidean nature of the unsupervised representations learned on various datasets. In the more specific area of CCMs, [1,11] propose hyperspherical variational autoencoders to better model data on a hypersphere, whereas [5,12] exploit CCMs with different curvatures to perform change detection on sequences of graphs. Several other works, however, show how different types of data can greatly benefit from being embedded on CCMs, with literature going as far back as [13], and the more recent notable contributions of [2] and [6].…”

“…The proposed solution uses adversarial learning to impose a soft constraint on the latent space of an autoencoder, training the network to produce a representation on the CCM. This work builds on the autoencoder model introduced in [5], which uses adversarial learning to map a stream of graphs on a CCM for performing change detection. Here, we give a novel formulation of the model that is not restricted to operate on sequences of graphs and show its effectiveness on different learning tasks not considered in [5].…”

“…This work builds on the autoencoder model introduced in [5], which uses adversarial learning to map a stream of graphs on a CCM for performing change detection. Here, we give a novel formulation of the model that is not restricted to operate on sequences of graphs and show its effectiveness on different learning tasks not considered in [5]. The model introduced here is characterised by a novel learning scheme that consists of optimising a combination of two objectives during the regu-larisation training phase of an adversarial autoencoder [14].…”