Hypergraph Motifs and Their Extensions Beyond Binary

TL;DR

This paper introduces hypergraph motifs and characteristic profiles to analyze the structure of real-world hypergraphs, enabling domain comparison, node classification, and hyperedge prediction with efficient algorithms and extended motifs.

Contribution

It defines hypergraph motifs and characteristic profiles, proposes parallel algorithms for counting motifs, and extends motifs to ternary hypergraphs, advancing hypergraph structural analysis.

Findings

H-motifs distinguish real-world hypergraphs from randomized ones.

Characteristic profiles capture domain-specific local structures.

Extended motifs improve node classification and hyperedge prediction.

Abstract

Hypergraphs naturally represent group interactions, which are omnipresent in many domains: collaborations of researchers, co-purchases of items, and joint interactions of proteins, to name a few. In this work, we propose tools for answering the following questions: (Q1) what are the structural design principles of real-world hypergraphs? (Q2) how can we compare local structures of hypergraphs of different sizes? (Q3) how can we identify domains from which hypergraphs are? We first define hypergraph motifs (h-motifs), which describe the overlapping patterns of three connected hyperedges. Then, we define the significance of each h-motif in a hypergraph as its occurrences relative to those in properly randomized hypergraphs. Lastly, we define the characteristic profile (CP) as the vector of the normalized significance of every h-motif. Regarding Q1, we find that h-motifs' occurrences in 11 real-world hypergraphs from 5 domains are clearly distinguished from those of randomized hypergraphs. Then, we demonstrate that CPs capture local structural patterns unique to each domain, and thus comparing CPs of hypergraphs addresses Q2 and Q3. The concept of CP is extended to represent the connectivity pattern of each node or hyperedge as a vector, which proves useful in node classification and hyperedge prediction. Our algorithmic contribution is to propose MoCHy, a family of parallel algorithms for counting h-motifs' occurrences in a hypergraph. We theoretically analyze their speed and accuracy and show empirically that the advanced approximate version MoCHy-A+ is more accurate and faster than the basic approximate and exact versions, respectively. Furthermore, we explore ternary hypergraph motifs that extends h-motifs by taking into account not only the presence but also the cardinality of intersections among hyperedges. This extension proves beneficial for all previously mentioned applications.