  We study the appearance of first-order dynamical phase transitions (DPTs) as
`intermittent' co-existing phases in the fluctuations of random walks on
graphs. We show that the diverging time scale leading to critical behaviour is
the waiting time to jump from one phase to another. This time scale is crucial
for observing the system's relaxation to stationarity and demonstrates
ergodicity of the system at criticality. We illustrate these results through
three analytical examples which provide insights into random walks exploring
random graphs.
