  Dynamical processes can be transformed into graphs through a family of
mappings called visibility algorithms, enabling the possibility of (i) making
empirical data analysis and signal processing and (ii) characterising classes
of dynamical systems and stochastic processes using the tools of graph theory.
Recent works show that the degree distribution of these graphs encapsulates
much information on the signals variability, and therefore constitutes a
fundamental feature for statistical learning purposes. However, exact solutions
for the degree distributions are only known in a few cases, such as for
uncorrelated random processes. Here we analytically explore these distributions
in a list of situations. We present a diagrammatic formalism which computes for
all degrees their corresponding probability as a series expansion in a coupling
constant which is the number of hidden variables. We offer a constructive
solution for general Markovian stochastic processes and deterministic maps. As
case tests we focus on Ornstein-Uhlenbeck processes, fully chaotic and
quasiperiodic maps. Whereas only for certain degree probabilities can all
diagrams be summed exactly, in the general case we show that the perturbation
theory converges. In a second part, we make use of a variational technique to
predict the complete degree distribution for special classes of Markovian
dynamics with fast-decaying correlations. In every case we compare the theory
with numerical experiments.
