The effect of the structure of a complex network on the properties of avalanche dynamical process on it has been analyzed for the first time. It has been shown that the assortativity (disassortativity) degree, which is a structure characteristic of the network and is numerically characterized by the assortativity coefficient r, is a control parameter governing the properties of the dynamical process on the network. The structure of individual avalanches on networks with various r values has been studied. It has been shown that the number of nodes involved in an avalanche is a periodic function of the time.
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