  We show that the recently introduced logarithmic metrics used to predict
disease arrival times on complex networks are approximations of more general
network-based measures derived from random walks theory. Using the daily
air-traffic transportation data we perform numerical experiments to compare the
infection arrival time with this alternative metric that is obtained by
accounting for multiple walks instead of only the most probable path. The
comparison with direct simulations of arrival times reveals a higher
correlation compared to the shortest path approach used previously. In addition
our method allows to connect fundamental observables in epidemic spreading with
the cumulant generating function of the hitting time for a Markov chain. Our
results provides a general and computationally efficient approach to the
problem using only algebraic methods.
