  Nature is a blossoming of regular structures, signature of self-organization
of the underlying microscopic interacting agents. Turing theory of pattern
formation is one of the most studied mechanisms to address such phenomena and
has been applied to a widespread gallery of disciplines. Turing himself used a
spatial discretization of the hosting support to eventually deal with a set of
ODEs. Such an idea contained the seeds of the theory on discrete support, which
has been fully acknowledged with the birth of network science in the early
2000s. This approach allows us to tackle several settings not displaying a
trivial continuous embedding, such as multiplex, temporal networks, and,
recently, higher-order structures. This line of research has been mostly
confined within the network science community, despite its inherent potential
to transcend the conventional boundaries of the PDE-based approach to Turing
patterns. Moreover, network topology allows for novel dynamics to be generated
via a universal formalism that can be readily extended to account for
higher-order structures. The interplay between continuous and discrete settings
can pave the way for further developments in the field.
