Temporally Evolving Generalised Networks

Networks have acquired increasing importance in both Machine Learning and Statistics communities. The language of metric graphs is very useful to generalise linear networks so that edges connecting vertices can be nonlinear. Further, a given process might be continuously observed over both edges and vertices. I shall illustrate the challenges related to the construction of generalised networks as well as their reproducing kernels. Then, I shall expose the problem of generalised networks that evolve over continuous or discrete time.
I shall put emphasis on two cases: in the first, a temporally evolving generalised framework is a product semi metric space. This implies that the topology of the graph is temporally invariant. In the second, we allow the topology of the graph to change over time. Hence, vertices can disappear (or new ones appear) and consequently edges connecting vertices will change. For both cases, building an appropriate semi metric is a challenging task. Then I shall illustrate the challenges related to isometric embeddings on suitable Hilbert spaces in order to build isotropic kernels on these networks.
Our findings are illustrated through a dataset on traffic accidents. We show an improved model performance based on a kernel specification and network-based metrics.

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