  Spreading processes are conventionally monitored on a macroscopic level by
counting the number of incidences over time. The spreading process can then be
modeled either on the microscopic level, assuming an underlying interaction
network, or directly on the macroscopic level, assuming that microscopic
contributions are negligible. The macroscopic characteristics of both
descriptions are commonly assumed to be identical. In this work, we show that
these characteristics of microscopic and macroscopic descriptions can be
different due to coalescence, i.e., a node being activated at the same time by
multiple sources. In particular, we consider a (microscopic) branching network
(probabilistic cellular automaton) with annealed connectivity disorder, record
the macroscopic activity, and then approximate this activity by a (macroscopic)
branching process. In this framework, we analytically calculate the effect of
coalescence on the collective dynamics. We show that coalescence leads to a
universal non-linear scaling function for the conditional expectation value of
successive network activity. This allows us to quantify the difference between
the microscopic model parameter and established macroscopic estimates. To
overcome this difference, we propose a non-linear estimator that correctly
infers the model branching parameter for all system sizes.
