STAND SPATIOTEMPORAL NONLOCAL AND NON-CONSERVATIVE DIFFUSION ON GRAPHS

Complex systems permeate biological life, society, and modern infrastructures. Their functioning depends primarily on the dynamic processes that take place between interacting entities that form network structures. Such structures can be appropriately described by mathematical models based on discrete systems, such as graphs and networks, and by dynamical systems supported by such networks. Among the latter, there is no doubt that diffusion plays a major role, being the main driver of dynamics ranging from information flow in social networks, transport in infrastructural systems, epidemic propagation processes and the synchronisation of biological, social, or technological entities. Classical diffusion is a conservative process in the graph that supports it. That is, the total number of diffusing particles at the vertices of the graph is constant over time. However, there are complex systems where non-conservative diffusion is ubiquitous. They include neural synapses, urban traffic, information diffusion in online social networks, trophic interactions in ecological systems, among others. Non-conservative (NC) diffusion has been previously studied in graphs, but it has not been developed enough to
occupy the important place it should have in the analysis of complex systems. Therefore, there is no mathematically plausible description in the current literature that encompasses the study of complex NC systems. Consequently, this basic research project foresees a significant advance in the knowledge of complex NC systems, developing a mathematical theory that will extend the knowledge of nonconservative diffusion to allow for non-locality in space and time, which will cover a very wide range of social, physical, biological, and technological processes occurring in the real world. This development is based on the elaboration of logistic reaction-diffusion models on graphs representing NC dynamics but reaching stationary equilibrium states. These models will be generalised to account for non-local effects in time and space. Spatial non-locality will be introduced by generalising the adjacency operator on graphs to the transformed d-path adjacency operators. Such transformations will be applied to account for the decay of spatial non-locality that takes place with the separation between vertices of the graph. On the other hand, temporal non-locality will be implemented by replacing the standard derivatives by Caputo time fractional derivatives in the NC diffusive models. Finally, the results will be integrated by studying the geometries induced by these non-local NC diffusive dynamics. For this purpose, the most probable trajectories of the diffusive particles will be studied through the geometry of the graphs. The models developed will be applied to the study of real-world problems such as chemical synapses in neural systems, urban traffic, and the diffusion of information in online social networks, for which the team has integrated international experts in its working team.