  There is great interest in predicting rare and extreme events in complex
systems, and in particular, understanding the role of network topology in
facilitating such events. In this work, we show that degree dispersion -- the
fact that the number of local connections in networks varies broadly --
increases the probability of large, rare fluctuations in population networks
generically. We perform explicit calculations for two canonical and distinct
classes of rare events: network extinction and switching. When the distance to
threshold is held constant, and hence stochastic effects are fairly compared
among networks, we show that there is a universal, exponential increase in the
rate of rare events proportional to the variance of a network's degree
distribution over its mean squared.
