The dynamics of pandemics is most often analyzed using a variation of the SIR (Susceptible-Infected-Recovered) model1, the key parameter of which is the basic reproduction number R0. Some evidences suggest that the contagion-spreading networks are scale-free, with the biggest nodes corresponding to superspreaders2,3. However, current understanding of the scale-free topology of these networks, and of the implications of such topology for the dynamics of pandemics is incomplete. Here we show that the world-wide spreading rate of COVID-19 gives an indirect evidence that the underlying virus-spreading network is scale-free, with the degree distribution exponent close to 2. Furthermore, our results show that the spreading rate of a virus is predominantly controlled by superspreaders who typically get infected and acquire immunity during the initial outbreak stage of the pandemic. Thereby the biggest nodes get immune and hence, removed from the network, resulting in a rapid decrease of the effective reproduction number. These findings are important for understanding the dynamics of pandemics, and for designing optimal virus control strategies. In particular, screening a population for the number of antibodies of a set of viruses can reveal potential superspreaders, the vaccination or isolation of whom can impede a pandemic at its early stage.