  Despite being a source of rich information, graphs are limited to pairwise
interactions. However, several real-world networks such as social networks,
neuronal networks, etc., involve interactions between more than two nodes.
Simplicial complexes provide a powerful mathematical framework to model such
higher-order interactions. It is well known that the spectrum of the graph
Laplacian is indicative of community structure, and this relation is exploited
by spectral clustering algorithms. Here we propose that the spectrum of the
Hodge Laplacian, a higher-order Laplacian defined on simplicial complexes,
encodes simplicial communities. We formulate an algorithm to extract simplicial
communities (of arbitrary dimension). We apply this algorithm to simplicial
complex benchmarks and to real higher-order network data including social
networks and networks extracted using language or text processing tools.
However, datasets of simplicial complexes are scarce, and for the vast majority
of datasets that may involve higher-order interactions, only the set of
pairwise interactions are available. Hence, we use known properties of the data
to infer the most likely higher-order interactions. In other words, we
introduce an inference method to predict the most likely simplicial complex
given the community structure of its network skeleton. This method identifies
as most likely the higher-order interactions inducing simplicial communities
that maximize the adjusted mutual information measured with respect to
ground-truth community structure. Finally, we consider higher-order networks
constructed through thresholding the edge weights of collaboration networks
(encoding only pairwise interactions) and provide an example of persistent
simplicial communities that are sustained over a wide range of the threshold.
