  We propose a method for obtaining parsimonious decompositions of networks
into higher order interactions which can take the form of arbitrary motifs.The
method is based on a class of analytically solvable generative models, where
vertices are connected via explicit copies of motifs, which in combination with
non-parametric priors allow us to infer higher order interactions from dyadic
graph data without any prior knowledge on the types or frequencies of such
interactions. Crucially, we also consider 'degree--corrected' models that
correctly reflect the degree distribution of the network and consequently prove
to be a better fit for many real world--networks compared to non-degree
corrected models. We test the presented approach on simulated data for which we
recover the set of underlying higher order interactions to a high degree of
accuracy. For empirical networks the method identifies concise sets of atomic
subgraphs from within thousands of candidates that cover a large fraction of
edges and include higher order interactions of known structural and functional
significance. The method not only produces an explicit higher order
representation of the network but also a fit of the network to analytically
tractable models opening new avenues for the systematic study of higher order
network structures.
