Fundamental dynamics of popularity-similarity trajectories in real networks

Real networks are complex dynamical systems, evolving over time with the addition and deletion of nodes and links. Currently, there exists no principled mathematical theory for their dynamics -- a grand-challenge open problem in complex networks. Here, we show that the popularity and similarity trajectories of nodes in hyperbolic embeddings of different real networks manifest universal self-similar properties with typical Hurst exponents $H \ll 0.5$. This means that the trajectories are anti-persistent or 'mean-reverting' with short-term memory, and they can be adequately captured by a fractional Brownian motion process. The observed behavior can be qualitatively reproduced in synthetic networks that possess a latent geometric space, but not in networks that lack such space, suggesting that the observed subdiffusive dynamics are inherently linked to the hidden geometry of real networks. These results set the foundations for rigorous mathematical machinery for describing and predicting real network dynamics

Systems of nonlinear PDEs arising in multilayer channel flows

This thesis presents analysis and computations of systems of nonlinear partial differential equations (PDEs) modelling the dynamics of three stratified immiscible viscous layers flowing inside a channel with parallel walls inclined to the horizontal. The three layers are separated by two fluid-fluid interfaces that are free to evolve spatiotemporally and nonlinearly when the flow becomes unstable. The determination of the flow involves solution of the Navier-Stokes in domains that are changing due to the evolution of the interfaces whose position must be determined as part of the solution, providing a hard nonlinear moving boundary problem. Long-wave approximation and a weakly nonlinear analysis of the Navier-stokes equations along with the associated boundary conditions, leads to reduced systems of nonlinear PDEs that in general form are systems of coupled Kuramoto- Sivashinsky equations. These physically derived coupled systems are mathematically rich due to the rather generic presence of coupled nonlinearities that undergo hyperbolic-elliptic transitions, along with high order dissipation. Analysis and numerical computations of the resulting coupled PDEs is presented in order to understand the stability of multilayer channel flows and explore and quantify the different types of underlying nonlinear phenomena that are crucial in applications. Importantly, it is found that multilayer flows can be unstable even at zero Reynolds numbers, in contrast to single interface problems. Furthermore, the thesis investigates the dynamical behaviour of the zero viscosity limits of the derived systems in order to verify their physical relevance as reduced models. Strong evidence of the existence of the zero viscosity limit is provided for mixed hyperbolic-elliptic type systems whose global existence is an open and challenging mathematical problem. Finally, a novel sufficient condition is derived for the occurrence of hyperbolic-elliptic transitions in general conservation laws of mixed type; the condition is demonstrated for several physical systems that have been studied in the literature.Open Acces