  Many networks must maintain synchrony despite the fact that they operate in
noisy environments. Important examples are stochastic inertial oscillators,
which are known to exhibit fluctuations with broad tails in many applications,
including electric power networks with renewable energy sources. Such
non-Gaussian fluctuations can result in rare network desynchronization. Here we
build a general theory for inertial oscillator network desynchronization by
non-Gaussian noise. We compute the rate of desynchronization and show that
higher-moments of noise enter at specific powers of coupling: either speeding
up or slowing down the rate exponentially depending on how noise statistics
match the statistics of a network's slowest mode. Finally, we use our theory to
introduce a technique that drastically reduces the effective description of
network desynchronization. Most interestingly, when instability is associated
with a single edge, the reduction is to one stochastic oscillator.
