Paris D. L. Flood

King Fai Yeh, Paris Flood, William Redman et al.

Recently, Koopman operator theory has become a powerful tool for developing linear representations of non-linear dynamical systems. However, existing data-driven applications of Koopman operator theory, including both traditional and deep learning approaches, perform poorly on non-linear network dynamics problems as they do not address the underlying geometric structure. In this paper we present a novel approach based on Koopman operator theory and message passing networks that finds a linear representation for the dynamical system which is globally valid at any time step. The linearisations found by our method produce predictions on a suite of network dynamics problems that are several orders of magnitude better than current state-of-the-art techniques. We also apply our approach to the highly non-linear training dynamics of neural network architectures, and obtain linear representations which can generate network parameters with comparable performance to networks trained by classical optimisers.

Paris D. L. Flood, Ramon Viñas, Pietro Liò

Link prediction in graphs is an important task in the fields of network science and machine learning. We investigate a flexible means of regularization for link prediction based on an approximation of the Kolmogorov complexity of graphs that is differentiable and compatible with recent advances in link prediction algorithms. Informally, the Kolmogorov complexity of an object is the length of the shortest computer program that produces the object. Complex networks are often generated, in part, by simple mechanisms; for example, many citation networks and social networks are approximately scale-free and can be explained by preferential attachment. A preference for predicting graphs with simpler generating mechanisms motivates our choice of Kolmogorov complexity as a regularization term. In our experiments the regularization method shows good performance on many diverse real-world networks, however we determine that this is likely due to an aggregation method rather than any actual estimation of Kolmogorov complexity.