  A unifying graph theoretic framework for the modelling of metro
transportation networks is proposed. This is achieved by first introducing a
basic graph framework for the modelling of the London underground system from a
diffusion law point of view. This forms a basis for the analysis of both
station importance and their vulnerability, whereby the concept of graph vertex
centrality plays a key role. We next explore k-edge augmentation of a graph
topology, and illustrate its usefulness both for improving the network
robustness and as a planning tool. Upon establishing the graph theoretic
attributes of the underlying graph topology, we proceed to introduce models for
processing data on such a metro graph. Commuter movement is shown to obey the
Fick's law of diffusion, where the graph Laplacian provides an analytical model
for the diffusion process of commuter population dynamics. Finally, we also
explore the application of modern deep learning models, such as graph neural
networks and hyper-graph neural networks, as general purpose models for the
modelling and forecasting of underground data, especially in the context of the
morning and evening rush hours. Comprehensive simulations including the
passenger in- and out-flows during the morning rush hour in London demonstrates
the advantages of the graph models in metro planning and traffic management, a
formal mathematical approach with wide economic implications.
