Synchronization of Complex Network Systems with Stochastic Disturbances

We study the robustness of the synchronization of coupled phase oscillators. When fluctuations of phase differences in lines caused by disturbance exceed a certain threshold, the state cannot return to synchrony thus leading to desynchronization. Our main result is the deviation of explicit formulas of a variance matrix that characterizes the severity of these fluctuations. We highlight the utility of these results in two general problems: vulnerable line identification and network design. We find that the vulnerability of lines can be encoded by the cycle space of graphs. It is analytically shown that a line in large-size cycles is more vulnerable than those in small-size cycles and adding a new line or increasing coupling strength of a line reduces the vulnerability of the lines in any cycle including this line, while it does not affect the vulnerability of the other lines.