  Visibility algorithms are a family of methods to map time series into
networks, with the aim of describing the structure of time series and their
underlying dynamical properties in graph-theoretical terms. Here we explore
some properties of both natural and horizontal visibility graphs associated to
several non-stationary processes, and we pay particular attention to their
capacity to assess time irreversibility. Non-stationary signals are
(infinitely) irreversible by definition (independently of whether the process
is Markovian or producing entropy at a positive rate), and thus the link
between entropy production and time series irreversibility has only been
explored in non-equilibrium stationary states. Here we show that the visibility
formalism naturally induces a new working definition of time irreversibility,
which allows to quantify several degrees of irreversibility for stationary and
non-stationary series, yielding finite values that can be used to efficiently
assess the presence of memory and off-equilibrium dynamics in non-stationary
processes without needs to differentiate or detrend them. We provide rigorous
results complemented by extensive numerical simulations on several classes of
stochastic processes.
