  Nonlinear dynamics play an important role in the analysis of signals. A
popular, readily interpretable nonlinear measure is Permutation Entropy. It has
recently been extended for the analysis of graph signals, thus providing a
framework for non-linear analysis of data sampled on irregular domains. Here,
we introduce a continuous version of Permutation Entropy, extend it to the
graph domain, and develop a ordinal activation function akin to the one of
neural networks. This is a step towards Ordinal Deep Learning, a potentially
effective and very recently posited concept. We also formally extend ordinal
contrasts to the graph domain. Continuous versions of ordinal contrasts of
length 3 are also introduced and their advantage is shown in experiments. We
also integrate specific contrasts for the analysis of images and show that it
generalizes well to the graph domain allowing a representation of images,
represented as graph signals, in a plane similar to the entropy-complexity one.
Applications to synthetic data, including fractal patterns and popular
non-linear maps, and real-life MRI data show the validity of these novel
extensions and potential benefits over the state of the art. By extending very
recent concepts related to permutation entropy to the graph domain, we expect
to accelerate the development of more graph-based entropy methods to enable
nonlinear analysis of a broader kind of data and establishing relationships
with emerging ideas in data science.
