Self-organization of a critical state on complex networks

The critical dynamics of a two-threshold system with the law of conservation of the basic quantity z and in the absence of sink on a scale-free network has been studied. It has been shown that the critical state that is a set of metastable states appears in such a system. The structure of the metastable states is a set of stable clusters of nodes at which the z values are close to the positive and negative threshold values. Avalanches transforming the system from one metastable state to another state appear in the system. The absence of sink is effectively replaced by the annihilation process. The statistics of avalanches in such a system has been analyzed. It has been shown that the self-organized critical state can appear in the system.