  The synchronization stability of a complex network system of coupled phase
oscillators is discussed. In case the network is affected by disturbances, a
stochastic linearized system of the coupled phase oscillators may be used to
determine the fluctuations of phase differences in the lines between the nodes
and to identify the vulnerable lines that may lead to desynchronization. The
main result is the derivation of the asymptotic variance matrix of the phase
differences which characterizes the severity of the fluctuations. It is found
that the cycle space of the graph of the system plays a role in this
characterization. With theory of the cycle space, the effect of forming small
cycles on the fluctuations are evaluated. It is proven that adding a new line
or increasing the coupling strength of a line affect the fluctuations in the
lines in any cycle including this line while it does not affect the
fluctuations in the other lines. In particular, if the phase differences at the
synchronous state are not changed by these actions, then the affected
fluctuations reduce.
