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

TL;DR

A novel change detection framework based on neural networks and CCMs, which takes into account the non-Euclidean nature of graphs is introduced, and the proposed methods are able to detect even small changes in a graph-generating process, consistently outperforming approaches based on Euclidean embeddings.

Abstract

The space of graphs is often characterised by a non-trivial 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 real-world 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.

Figures (8)

• Figure 1: Schematic representation of the exp-map and log-map for $\mathcal{M}_1$, with a tangent space $T_x\mathcal{M}_1$ at $x$. We adopt a representation of the tangent plane such that the origin $x \in \mathcal{M}_\kappa$ of the log-map is mapped to the origin of the tangent plane (denoted $0$ in the figure) and vice versa with exp-map bridson2013metric.
• Figure 2: Schematic view of the AAE with a spherical CCM, in the probabilistic setting. From left to right, top to bottom: the AAE takes as input graphs represented by their adjacency matrix $A$, node features $X$, and edge attributes $E$, and outputs reconstructions of the same three matrices $\hat{A}$, $\hat{X}$, and $\hat{E}$ (blue path). The discriminator is trained to distinguish between samples produced by the encoder and samples coming from the true prior (yellow path). Finally, the encoder is updated to fool the discriminator by maximising its classification error. The encoder consists of graph convolutional layers, followed by a global pooling layer to obtain a graph embedding in the ambient space ($\mathbb{R}^{d+1}$). Embeddings are then constrained to the CCM $\mathcal{M}_k$ by 1) matching the prior $\mathcal{P}_{\mathcal{M}_\kappa}(\theta)$, and 2) orthogonally projecting the points onto the CCM. The decoder is a dense network with three parallel outputs for $\hat{A}$, $\hat{X}$, and $\hat{E}$. The discriminator is a dense network with sigmoid output. Best viewed in colour.
• Figure 3: Example of Delaunay triangulations of classes 0 to 4. The same colours are used in Figure \ref{['fig:embeddings_delaunay']} to represent the embeddings on the CCMs.
• Figure 4: Hammer and planar projections of embeddings produced by AAEs with latent CCMs $\mathcal{M}_1$ and $\mathcal{M}_{-1}$, respectively, for Delaunay triangulations of classes 0 to 4 (c.f. Figure \ref{['fig:delaunay_graphs']}). Best viewed in colour.
• Figure 5: Example of functional connectivity network extracted from a 1-second clip of iEEG data, using Pearson's correlation. Only edge attributes are shown in the figure.