Retrieving Top-k Hyperedge Triplets: Models and Applications

TL;DR

It is argued that the relative sizes of hyperedge intersections within motifs contain varied and valuable information, and a suite of efficient algorithms for finding top-k triplets of hyperedges based on optimizing the sizes of these intersection patterns are proposed.

Abstract

Complex systems frequently exhibit multi-way, rather than pairwise, interactions. These group interactions cannot be faithfully modeled as collections of pairwise interactions using graphs and instead require hypergraphs. However, methods that analyze hypergraphs directly, rather than via lossy graph reductions, remain limited. Hypergraph motifs hold promise in this regard, as motif patterns serve as building blocks for larger group interactions which are inexpressible by graphs. Recent work has focused on categorizing and counting hypergraph motifs based on the existence of nodes in hyperedge intersection regions. Here, we argue that the relative sizes of hyperedge intersections within motifs contain varied and valuable information. We propose a suite of efficient algorithms for finding top-k triplets of hyperedges based on optimizing the sizes of these intersection patterns. This formulation uncovers interesting local patterns of interaction, finding hyperedge triplets that either (1) are the least similar with each other, (2) have the highest pairwise but not groupwise correlation, or (3) are the most similar with each other. We formalize this as a combinatorial optimization problem and design efficient algorithms based on filtering hyperedges. Our comprehensive experimental evaluation shows that the resulting hyperedge triplets yield insightful information on real-world hypergraphs. Our approach is also orders of magnitude faster than a naive baseline implementation.

Figures (7)

• Figure 1: A few examples of $h$-motifs, as denoted in lee2020hypergraph. Each circle denotes an hyperedge and intersections represent the set of common nodes. Colored-regions are non-empty.
• Figure 2: Closed $h$-motif counts for Gene-Disease. The x-axis contains arbitrary motif numberings for the closed $h$-motifs in lee2020hypergraph. Figure \ref{['fig:hmotifs']} shows the top four $h$-motifs in the original network.
• Figure 3: Hyperedge triplet regions. Figure \ref{['fig:regionLabels']} depicts the independent ($R_1$), disjoint ($R_2$), and common ($R_3$) regions for the hyperedge triplet $\{A, B, C\}$. Figure \ref{['fig:weightExample']} shows a toy triplet where the independent weight is $W_1(T)=\frac{\min(7, 5, 6)}{1+(2+2+3+1)}=\frac{5}{9}$, the disjoint weight is $W_2(T)=\frac{\min(2, 2, 3)}{1+1}=1$, and the common weight is $W_3(T)=\frac{\min(1)}{1}=1$.
• Figure 4: Relative entropies of the independent and disjoint regions for the top 1M independent and disjoint triplets, respectively. Entropies are between the colored regions in the legend.
• Figure 5: The maximum disjoint hyperedge triplet in Yelp: Ruby Slipper Cafe, Coop's Place, and The Original Pierre Maspero's in New Orleans.