Controlling the spreading in small-world evolving networks: stability, oscillation, and topology

The spreading of viruses, diseases, and even disasters (such as power blackouts and financial crises) in many large-scale and small-world networks is one of the mostly concerned issues today. In this note, we study general spreading dynamical behaviors in small-world evolving networks when control strategies are applied to suppress the propagation of diseases, viruses, and disasters. After proposing a novel Watts-Strogatz (W-S) spreading model to capture the general spreading mechanism in small-world networks, we investigate the stability and Hopf bifurcations of delay-controlled spreading models with linear and nonlinear feedback controllers, where parameters of small-world rewiring probability, feedback control gain, and time delay are analyzed for the oscillating behaviors. We conclude that the oscillatory spreading phenomena in delay-controlled small-world networks are topologically inherent.