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The synchronization stability of a complex network system of coupled phaseoscillators is discussed. In case the network is affected by disturbances, astochastic linearized system of the coupled phase oscillators may be used todetermine the fluctuations of phase differences in the lines between the nodesand to identify the vulnerable lines that may lead to desynchronization. Themain result is the derivation of the asymptotic variance matrix of the phasedifferences which characterizes the severity of the fluctuations. It is foundthat the cycle space of the graph of the system plays a role in thischaracterization. With theory of the cycle space, the effect of forming smallcycles on the fluctuations are evaluated. It is proven that adding a new lineor increasing the coupling strength of a line affect the fluctuations in thelines in any cycle including this line while it does not affect thefluctuations in the other lines. In particular, if the phase differences at thesynchronous state are not changed by these actions, then the affectedfluctuations reduce.

Synchronization of Coupled Phase Oscillators with Stochastic Disturbances and the Cycle Space of the Graph