  We study the probability distribution of age of information (AoI) in
arbitrary networks with memoryless service times. A source node generates
packets following a Poisson process, and then the packets are forwarded across
the network in such a way that newer updates preempt older ones. This model is
equivalent to gossip networks that was recently studied by Yates, and for which
he obtained a recursive formula allowing the computation for the average AoI.
In this paper, we obtain a very simple characterization of the stationary
distribution of AoI at every node in the network. This allows for the
computation of the average of an arbitrary function of the age. In particular,
we can compute age-violation probabilities. Furthermore, we show how it is
possible to use insights from our simple characterization in order to
substantially reduce the computation time of average AoIs in some structured
networks. Finally, we describe how it is possible to use our characterization
in order to obtain faster and more accurate Monte Carlo simulations estimating
the average AoI, or the average of an arbitrary function of the age.
