Continuous-Time Random Walks and Temporal Networks
Real-world networks often exhibit complex temporal patterns that affect their dynamics and function. In this chapter, we focus on the mathematical modelling of diffusion on temporal networks, and on its connection with continuous-time random walks. In that case, it is important to distinguish active walkers, whose motion triggers the activity of the network, from passive walkers, whose motion is restricted by the activity of the network. One can then develop renewal processes for the dynamics of the walker and for the dynamics of the network respectively, and identify how the shape of the temporal distribution affects spreading. As we show, the system exhibits non-Markovian features when the renewal process departs from a Poisson process, and different mechanisms tend to slow down the exploration of the network when the temporal distribution presents a fat tail. We further highlight how some of these ideas could be generalised, for instance to the case of more general spreading processes.
KeywordsTemporal networks Random walks Diffusion Renewal process Mixing
I would like to thank my many collaborators without whom none of this work would have been done and, in particular, Naoki Masuda for co-writing  that was a great inspiration for this chapter.
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