  Spreading on networks is influenced by a number of factors including
different parts of the inter-event time distribution (IETD), the topology of
the network and non-stationarity. In order to understand the role of these
factors we study the SI model on temporal networks with different aggregated
topologies and different IETDs. Based on analytic calculations and numerical
simulations, we show that if the stationary bursty process is governed by
power-law IETD, the spreading can be slowed down or accelerated as compared to
a Poisson process; the speed is determined by the short time behaviour, which
in our model is controlled by the exponent. We demonstrate that finite, so
called "locally tree-like" networks, like the Barab\'asi-Albert networks behave
very differently from real tree graphs if the IETD is strongly fat-tailed, as
the lack or presence of rare alternative paths modifies the spreading. A
further important result is that the non-stationarity of the dynamics has a
significant effect on the spreading speed for strongly fat-tailed power-law
IETDs, thus bursty processes characterized by small power-law exponents can
cause slow spreading in the stationary state but also very rapid spreading
heavily depending on the age of the processes.
