Nonparametric inference of higher order interaction patterns in networks

TL;DR

The paper tackles inferring higher-order network interactions beyond dyadic edges via a nonparametric Bayesian framework that treats motifs as explicit atoms. It builds subgraph configurations (atoms) and uses degree-corrected microcanonical SGCMs to define likelihoods, and seeks the MAP configuration $C$ by maximizing $P(C|G)$ with $P(G|C)=1$ whenever $\bigcup_{s\in C} E(s)=E(G)$. Model comparison relies on posterior odds and minimum description length (MDL) with $\Sigma(C)=S(C|\mathbf{d},\mathbf{n_m},M)+\epsilon(\mathbf{d},\mathbf{n_m},M)$, enabling selection among atom sets and degree-structure variants. Across real networks—the Malaria Genes network, network-science co-authorship, the directed connectome, E. coli metabolism, and the C. elegans synaptic network—yields compact MAP configurations that recover known higher-order patterns such as triangles, 4-cycles, and feed-forward loops.

Abstract

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.

Figures (3)

• Figure 1: A subgraph configuration consisting two single edges, a triangle, a 4-cycle and a diamond.
• Figure 2: Examples of undirected and directed motifs with vertex colours indicating orbits.
• Figure 3: Visualization of the largest connected component of collaborations between network scientists. Colours represent different atomic subgraph types in the MAP-configuration. A high resolution plot of the full network can be found in the SI.