  Crystals arise as the result of the breaking of a spatial translation
symmetry. Similarly, translation symmetries can also be broken in time so that
discrete time crystals appear. Here, we introduce a method to describe,
characterize, and explore the physical phenomena related to this phase of
matter using tools from graph theory. The analysis of the graphs allows to
visualizing time-crystalline order and to analyze features of the quantum
system. For example, we explore in detail the melting process of a minimal
model of a period-2 discrete time crystal and describe it in terms of the
evolution of the associated graph structure. We show that during the melting
process, the network evolution exhibits an emergent preferential attachment
mechanism, directly associated with the existence of scale-free networks. Thus,
our strategy allows us to propose a previously unexplored far-reaching
application of time crystals as a quantum simulator of complex quantum
networks.
