  We investigate how the properties of inhomogeneous patterns of activity,
appearing in many natural and social phenomena, depend on the temporal
resolution used to define individual bursts of activity. To this end, we
consider time series of microscopic events produced by a self-exciting Hawkes
process, and leverage a percolation framework to study the formation of
macroscopic bursts of activity as a function of the resolution parameter. We
find that the very same process may result in different distributions of
avalanche size and duration, which are understood in terms of the competition
between the 1D percolation and the branching process universality class. Pure
regimes for the individual classes are observed at specific values of the
resolution parameter corresponding to the critical points of the percolation
diagram. A regime of crossover characterized by a mixture of the two universal
behaviors is observed in a wide region of the diagram. The hybrid scaling
appears to be a likely outcome for an analysis of the time series based on a
reasonably chosen, but not precisely adjusted, value of the resolution
parameter.
