Do We Need an Encoder-Decoder to Model Dynamical Systems on Networks?
This addresses a critical misunderstanding in modeling dynamical systems on networks, potentially improving accuracy for researchers in network science and machine learning, though it is incremental as it builds on existing methods.
The paper identifies that latent vertex embeddings in neural dynamical network models, while fitting observations well, lead to incorrect long-term dynamical behaviors, and proposes an embedding-free alternative that reliably recovers a broad class of dynamics on networks from time series data.
As deep learning gains popularity in modelling dynamical systems, we expose an underappreciated misunderstanding relevant to modelling dynamics on networks. Strongly influenced by graph neural networks, latent vertex embeddings are naturally adopted in many neural dynamical network models. However, we show that embeddings tend to induce a model that fits observations well but simultaneously has incorrect dynamical behaviours. Recognising that previous studies narrowly focus on short-term predictions during the transient phase of a flow, we propose three tests for correct long-term behaviour, and illustrate how an embedding-based dynamical model fails these tests, and analyse the causes, particularly through the lens of topological conjugacy. In doing so, we show that the difficulties can be avoided by not using embedding. We propose a simple embedding-free alternative based on parametrising two additive vector-field components. Through extensive experiments, we verify that the proposed model can reliably recover a broad class of dynamics on different network topologies from time series data.