Spreading on networks is influenced by a number of factors includingdifferent parts of the inter-event time distribution (IETD), the topology ofthe network and non-stationarity. In order to understand the role of thesefactors we study the SI model on temporal networks with different aggregatedtopologies and different IETDs. Based on analytic calculations and numericalsimulations, we show that if the stationary bursty process is governed bypower-law IETD, the spreading can be slowed down or accelerated as compared toa Poisson process; the speed is determined by the short time behaviour, whichin our model is controlled by the exponent. We demonstrate that finite, socalled "locally tree-like" networks, like the Barab\'asi-Albert networks behavevery differently from real tree graphs if the IETD is strongly fat-tailed, asthe lack or presence of rare alternative paths modifies the spreading. Afurther important result is that the non-stationarity of the dynamics has asignificant effect on the spreading speed for strongly fat-tailed power-lawIETDs, thus bursty processes characterized by small power-law exponents cancause slow spreading in the stationary state but also very rapid spreadingheavily depending on the age of the processes.
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