  A simplicial complex is a generalization of a graph: a collection of n-ary
relationships (instead of binary as the edges of a graph), named simplices. In
this paper, we develop a new tool to study the structure of simplicial
complexes: we generalize the graph notion of truss decomposition to complexes,
and show that this more powerful representation gives rise to different
properties compared to the graph-based one. This power, however, comes with
important computational challenges derived from the combinatorial explosion
caused by the downward closure property of complexes. Drawing upon ideas from
itemset mining and similarity search, we design a memory-aware algorithm,
dubbed STruD, which is able to efficiently compute the truss decomposition of a
simplicial complex. STruD adapts its behavior to the amount of available memory
by storing intermediate data in a compact way. We then devise a variant that
computes directly the n simplices of maximum trussness. By applying STruD to
several datasets, we prove its scalability, and provide an analysis of their
structure. Finally, we show that the truss decomposition can be seen as a
filtration, and as such it can be used to study the persistent homology of a
dataset, a method for computing topological features at different spatial
resolutions, prominent in Topological Data Analysis.
