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.

翻译：我们提出一种方法，用于将网络简洁分解为可呈现任意模体形式的高阶交互。该方法基于一类可解析求解的生成模型，其中顶点通过显式复制的模体相连，结合非参数先验，使我们能够在缺乏交互类型或频率先验知识的情况下，从二元图数据推断高阶交互。关键的是，我们还考虑了"度校正"模型，该模型能正确反映网络的度分布，与未度校正模型相比，对许多真实网络拟合更优。我们在模拟数据上测试了该方法，能以高精度恢复底层高阶交互集合。对于经验网络，该方法能从数千个候选子图中识别出简洁的原子子图集，这些子图覆盖大量边，并包含已知结构和功能意义的高阶交互。该方法不仅提供网络的显式高阶表示，还将网络拟合至解析可解的模型，为高阶网络结构的系统研究开辟新途径。