The role of directed cycles in a directed neural network

This paper investigates the dynamics of a directed acyclic neural network by edge adding control. We find that the local stability and Hopf bifurcation of the controlled network only depend on the size and intersection of directed circles, instead of the number and position of the added edges. More specifically, if there is no circle in the controlled network, the local dynamics of the network will remain unchanged and Hopf bifurcation will not occur even if the number of added edges is sufficient. However, if there exist circles, then the network may undergo Hopf bifurcation. Our results show that the circle structure is a necessary condition for the generation of Hopf bifurcation, and the bifurcation threshold is determined by the number, size, and intersection of circles. Numerical experiments are provided to support the validity of the theory.