Recently, there has been an increasing number of empirical evidence supporting the hypothesis that spread of avalanches of microposts on social networks, such as Twitter, is associated with some sociopolitical events. Typical examples of such events are political elections and protest movements. Inspired by this phenomenon, we built a phenomenological model that describes Twitter’s self‐organization in a critical state. An external manifestation of this condition is the spread of avalanches of microposts on the network. The model is based on a fractional three‐parameter self‐organization scheme with stochastic sources. It is shown that the adiabatic mode of self‐organization in a critical state is determined by the intensive coordinated action of a relatively small number of network users. To identify the critical states of the network and to verify the model, we have proposed a spectrum of three scaling indicators of the observed time series of microposts.