  The richness of many complex systems stems from the interactions among their
components. The higher-order nature of these interactions, involving many units
at once, and their temporal dynamics constitute crucial properties that shape
the behaviour of the system itself. An adequate description of these systems is
offered by temporal hypergraphs, that integrate these features within the same
framework. However, tools for their temporal and topological characterization
are still scarce. Here we develop a series of methods specifically designed to
analyse the structural properties of temporal hypergraphs at multiple scales.
Leveraging the hyper-core decomposition of hypergraphs, we follow the evolution
of the hyper-cores through time, characterizing the hypergraph structure and
its temporal dynamics at different topological scales, and quantifying the
multi-scale structural stability of the system. We also define two static
hypercoreness centrality measures that provide an overall description of the
nodes aggregated structural behaviour. We apply the characterization methods to
several data sets, establishing connections between structural properties and
specific activities within the systems. Finally, we show how the proposed
method can be used as a model-validation tool for synthetic temporal
hypergraphs, distinguishing the higher-order structures and dynamics generated
by different models from the empirical ones, and thus identifying the essential
model mechanisms to reproduce the empirical hypergraph structure and evolution.
Our work opens several research directions, from the understanding of dynamic
processes on temporal higher-order networks to the design of new models of
time-varying hypergraphs.
