FRONT PROPAGATION AND ARRIVAL TIMES IN NETWORKS WITH APPLICATION TO NEURODEGENERATIVE DISEASES

Many physical, epidemiological, or physiological dynamical processes on networks support front-like propagation, where an initial localized perturbation grows and systematically invades all nodes in the network. A key problem is then to extract estimates for the dynamics. In particular, if a single node is seeded at a small concentration, when will other nodes reach the same initial concentration? Here, motivated by the study of toxic protein propagation in neurodegenerative diseases, we present and compare three different estimates for the arrival time in order of increasing analytical complexity: the linear arrival time, obtained by linearizing the underlying dynamical system; the Lambert time, obtained by considering the interaction of pairs of nodes; and the nonlinear arrival time, obtained by asymptotic techniques. We use the classic Fisher-Kolmogorov-PetrovskyPiskunov equation as a paradigm for the dynamics and show that each method provides different insights but consistent time estimates. Further, we show that the nonlinear asymptotic method also gives an approximate solution, valid in the entire domain, and the correct ordering of arrival regions over large regions of parameters and initial conditions.

Automated summary

Many dynamical processes on networks exhibit front-like propagation, and the problem of estimating the dynamics is important. In this study, we consider toxic protein propagation in neurodegenerative diseases and compare different methods for estimating the arrival time, including linear, Lambert, and nonlinear approaches. We use the Fisher-Kolmogorov-Petrovsky-Piskunov equation as a model and show that each method provides valuable insights and consistent time estimates.