  A mapping of avalanches occurring in the zero-temperature random-field Ising
model (zt-RFIM) to life-periods of a population experiencing immigration is
established. Such a mapping allows the microscopic criteria for occurrence of
an infinite avalanche in a q-regular graph to be determined. A key factor for
an avalanche of spin flips to become infinite is that it interacts in an
optimal way with previously flipped spins. Based on these criteria, we explain
why an infinite avalanche can occur in q-regular graphs only for q>3, and
suggest that this criterion might be relevant for other systems. The generating
function techniques developed for branching processes are applied to obtain
analytical expressions for the duration, pulse-shapes and power spectrum of the
avalanches. The results show that only very long avalanches exhibit a
significant degree of universality.
