Transcritical bifurcation yielding global stability for network processes

Abstract

Several network processes exhibiting transcritical bifurcation have globally stable steady states. Their dynamical behaviour is captured by a simple property of the right hand side of the corresponding system of ODEs. Based on this property, a class of dynamical systems is introduced, for which the local stability of the trivial steady state determines the global behaviour of the system. It is shown that this condition is satisfied by three network models, namely the individual-based and degree-based ODE approximations of SIS epidemic propagation on networks and the Hopfield model with non-negative weights. The general result enables us to describe the global behaviour of these systems that was not available for the first and third models and was proved in a significantly more complicated way for the second.