  Time-limited states characterise many dynamical processes on networks:
disease infected individuals recover after some time, people forget news
spreading on social networks, or passengers may not wait forever for a
connection. These dynamics can be described as limited waiting-time processes,
and they are particularly important for systems modelled as temporal networks.
These processes have been studied via simulations, which is equivalent to
repeatedly finding all limited-waiting time temporal paths from a source node
and time. We propose a method yielding orders of magnitude more efficient way
of tracking the reachability of such temporal paths. Our method gives
simultaneous estimates of the in- or out-reachability (with any chosen
waiting-time limit) from every possible starting point and time. It works on
very large temporal networks with hundreds of millions of events on current
commodity computing hardware. This opens up the possibility to analyse
reachability and dynamics of spreading processes on large temporal networks in
completely new ways. For example, one can now compute centralities based on
global reachability for all events or can find with high probability the
infected node and time, which would lead to the largest epidemic outbreak.
