Foundations of diffusion and instabilities in nonlinear evolution equations on temporal graphs and graphons

Student thesis: Doc typesDoctor of Sciences

Abstract

Evolution equations on graphs describe the development over time of a system on a domain made of nodes and links, starting from an initial state. They apply to phenomena so diverse that they will likely never develop into a single coherent field. However, a central question is to reveal the impact of the structural properties of the graph on the trajectory of the system.

This thesis is an attempt to further uncover this interplay between structure and dynamics, beyond the simple paradigm of static networks of moderate size. Empirical evidence indeed shows that some networks possess inherent temporal properties. On another note, massive graphs have become commonplace in real-life and scientific research, and a range of graph-theoretical methods and algorithms face scalability issues, demanding to consider graphs as if they were continuous objects.

We study two classes of problems: linear diffusion and then nonlinear variants, where local reactions combine with some form of diffusion through the edges of the graph. We revisit well-known stability-instability properties for such systems, and reveal significant effects due to the temporal nature of the graph. As a side note, we examine the inclusion of delay in the diffusive process, as a reasonable way to improve the models and refine the match with observations, in systems where the time for communication, reaction or decision-making cannot be neglected.

We also prove the validity of the continuum-limit approach to random walks on dense weighted graphs, in discrete and in continuous time, relying on the framework provided by graph-limit theory. We answer positively to the question of transferability, showing that the cost of duplicating the analysis for each finite graph can be avoided by considering instead the continuum problem. The dissertation ends with a study of diffusion-driven instabilities in reaction-diffusion systems on graphons which are the continuum-limit version of graphs.
Date of Award25 Jun 2020
Original languageEnglish
Awarding Institution
  • University of Namur
SponsorsEcole Royale Militaire
SupervisorTimoteo Carletti (Supervisor), Ben Lauwens (Supervisor), ALEXANDRE MAUROY (President), Duccio Fanelli (Jury), Hiroya Nakao (Jury) & Johan Gallant (Jury)

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